JHEP03(2013)098

Published for SISSA by Springer

Received: January 3, 2013

Accepted: March 3, 2013

Published: March 18, 2013

U(1) symmetries in F-theory GUTs with multiple

sections

Christoph Mayrhofer,a Eran Paltib and Timo Weiganda

aInstitut fur Theoretische Physik, Universitat Heidelberg,

Philosophenweg 19, D-69120 Heidelberg, GermanybCentre de Physique Theorique, Ecole Polytechnique,

CNRS, Palaiseau, France

E-mail: c.mayrhofer@thphys.uni-heidelberg.de,

t.weigand@thphys.uni-heidelberg.de, eran.palti@cpht.polytechnique.fr

Abstract: We present a systematic construction of F-theory compactifications with

Abelian gauge symmetries in addition to a non-Abelian gauge group G. The formalism is

generally applicable to models in global Tate form but we focus on the phenomenologically

interesting case of G = SU(5). The Abelian gauge factors arise due to extra global sections

resulting from a specific factorisation of the Tate polynomial which describes the elliptic

fibration. These constructions, which accommodate up to four different U(1) factors, are

worked out in detail for the two possible embeddings of a single U(1) factor into E8, usu-

ally denoted SU(5)×U(1)X and SU(5)×U(1)PQ. The resolved models can be understood

either patchwise via a small resolution or in terms of a P1,1,2[4] description of the elliptic

fibration. We derive the U(1) charges of the fields from the geometry, construct the U(1)

gauge fluxes and exemplify the structure of the Yukawa interaction points. A particularly

interesting result is that the global SU(5) × U(1)PQ model exhibits extra SU(5)-singlet

states which are incompatible with a single global decomposition of the 248 of E8. The

states in turn lead to new Yukawa type couplings which have not been considered in local

model building.

Keywords: F-Theory, Gauge Symmetry, GUT

Open Access doi:10.1007/JHEP03(2013)098

JHEP03(2013)098

Contents

1 Introduction 1

2 U(1) symmetries from the factorised Tate model 5

2.1 Engineering extra sections by factorisation 5

2.2 Resolving the SU(5) singularity 8

2.3 Factorised Tate models 10

2.3.1 4− 1 factorisation 11

2.3.2 3− 2 factorisation 12

3 Fibre structure and charges in the U(1)X model 13

3.1 Structure of the matter surfaces 13

3.2 U(1)X charges for matter curves and G4-flux 18

3.3 Yukawa points 20

4 Fibre structure and charges in the U(1)PQ model 21

5 Factorised SU(5) × U(1) models as P[1,1,2]-fibrations 26

5.1 The SU(5) resolution 27

5.2 The singlet curves in the resolved P[1,1,2]-fibration 30

6 Summary 32

A More factorised Tate models 33

A.1 2− 2− 1 factorisation 33

A.2 3− 1− 1 factorisation 35

A.3 2− 1− 1− 1 factorisation 36

A.4 1− 1− 1− 1− 1 factorisation 37

B Relation to other approaches to U(1)s 38

B.1 Relation to the U(1)-restricted Tate model 38

B.2 Relation to split spectral cover models 39

1 Introduction

The exceptional group E8 plays a central role in string theory and in particular in the

Heterotic string and F-theory. Within the context of F-theory Grand Unified Theories

(GUTs) [1–4] an underlying E8 implies that the GUT group, minimally SU(5), is natu-

rally extended by additional symmetries coming from its embedding in E8. Particularly

interesting for model building are additional U(1) symmetries. These play two important

– 1 –

JHEP03(2013)098

roles: they can support gauge flux thereby inducing chirality in the massless spectrum,

and they can lead to gauge and global symmetries beyond the Standard Model that can

be used to control the structure of the low energy theory, e.g. to forbid proton decay or

generating flavour hierarchies. In F-theory [5–9] gauge symmetries are geometric in origin

and so understanding the geometry associated to U(1) symmetries is crucial to this aspect

of model building. Indeed a lot of recent progress has been made towards understanding

the explicit construction and fully global aspects of U(1)s in four-dimensional F-theory

compactifications [10–16] and in compactifications to six dimensions [17]. In the local ap-

proach to F-theory U(1) symmetries are quite well understood within the spectral cover

approach [18] and have been used extensively in local model building [19–37]. However,

many important aspects of U(1) symmetries are inherently global in nature: they can be

broken away from the GUT brane [10, 38] and the associated gauge flux is not localised on

the GUT brane. Therefore a global understanding of U(1) symmetries is one of the central

requirements for realistic F-theory model building.

In the weakly coupled type IIB limit it is possible to study intersecting D7-brane

configurations generally without specifying the explicit geometry of the Calabi-Yau three-

fold. The analogous general procedure for F-theory models is the study of the elliptic

fibration without specifying the base explicitly. The form that the elliptic fibration must

take in order to induce a non-Abelian singularity, or equivalently gauge group, over a divisor

in the base is very well understood and given a fibration the non-Abelian structure can be

discerned in an algorithmic way [39, 40]. Further, the U(1) components that make up the

Cartan of a non-Abelian singularity can be studied explicitly by considering the M-theory

dual. On the M-theory side it is possible to resolve the non-Abelian singularity, which in

the gauge theory corresponds to moving along the Coulomb branch. After resolving the

singularity each Cartan element corresponds to a resolution divisor whose dual two-form

gives rise to a U(1) gauge field from dimensional reduction of the M-theory three-form C3.

In the context of four-dimensional F-theory SU(5) GUT models this procedure has been

carried out, using various techniques, in [12–14, 41–44] (see [45–50] for other gauge groups).

Abelian symmetries that are not in the Cartan of a non-Abelian singularity are less well

understood. The first complication is that in string theory such isolated U(1) symmetries

can often gain a Stuckelberg mass removing them from the massless spectrum. In the

weakly coupled IIB limit this is possible even in the absence of any flux, and such a purely

geometrically massive U(1) would be very difficult to identify in the F-theory uplift. Some

progress towards understanding such U(1)s was made in [10, 51] where they were proposed

to uplift to non-closed two-forms on the M-theory side. Further in [14] the flux associated

to one such massive U(1), the diagonal one in the IIB limit, was identified. However a

general procedure for identifying and constructing such U(1) symmetries is missing and we

will have nothing new to say regarding them in this paper.

A more tractable class of U(1) symmetries are those which remain massless in the

absence of any flux. A general approach to such U(1) symmetries should involve the

construction of multiple sections of the elliptic fibration. It was shown already in [8, 9] that

isolated U(1)s correspond to additional sections beyond the universal one which specifies the

embedding of the base. Concrete investigations of U(1) symmetries in six-dimensional F-

– 2 –

JHEP03(2013)098

theory compactifications have appeared early on in [52–54]. In the context of SU(5) GUTs

one approach to realising massless U(1)s was proposed in [10] in terms of what was called an

U(1)-restricted Tate model. The idea was to choose the coefficients of the elliptic fibration,

which already have an SU(5) singularity over the GUT divisor, so as to induce an additional

SU(2) singularity in the fiber over a curve in the base. The resolution of this singularity

introduces a new divisor which is associated to the additional U(1) symmetry [10, 12–14].

In [11] it was shown that the SU(2) singularity can be written in the form of a conifold

which is then resolved. The same procedure was used [15] to construct models with an

additional U(1) symmetry. The U(1)-restricted Tate model is quite well understood by

now. However this model realises only one particular embedding of a single additional

U(1) symmetry in E8 and additionally strongly restricts the possible matter spectrum, by

turning off one of the 10-matter curves, from the most general configuration. This is in

contrast to the rich structure of U(1) symmetries and matter spectra that are possible in

breaking E8 → SU(5) which have been used in local model building [19–37]. Indeed in its

local limit, i.e. in the projection to the SU(5) GUT divisor, the U(1) restricted Tate model

flows to the SU(5)×U(1)X split spectral cover [18, 19]. Locally, one usually thinks of the

gauge group SU(5) as arising from an underlying E8 symmetry and the possible U(1)s then

arise from the various embeddings into E8 [22, 24]. The purpose of this paper is to study

how the full spectrum of possibilities can be realised in a global setting thereby opening

the way to realising the phenomenology of local models in a global string vacuum. This

route will offer some surprises.

The key idea of our present paper is that we construct Tate models which give in a

specific way multiple sections. We call these factorised Tate models. We will show that

such models automatically induce a binomial singularity on the manifold whose resolution

gives rise to the appropriate U(1)s and their fluxes. The importance of suitable global

factorisations of the spectral cover equation to construct heterotic models with (multiple)

U(1) symmetries has been recently explored in [55]. Our approach is independent of any

heterotic dual.

By explicitly resolving and studying the fibre structure in detail for some examples we

derive the matter spectrum and the associated U(1) charges directly from the geometry. A

crucial aspect of the construction is that, unlike in the U(1)-restricted model, we will recover

the full matter spectrum with no additional constraints on the matter curves. We work

out the details of these matter curves and their resolved fiber structure both for models

with so-called U(1)X charge and with U(1)PQ charge. The latter has been used intensively

in local model building because the Peccei-Quinn symmetry can solve the µ-problem and

forbid dimension-five proton decay operators [20, 21], and our analysis provides the first

global embedding of this scenario.

The spectrum of SU(5) charged matter indeed assembles into representations that

can be obtained by the decomposition of the adjoint of one E8 into SU(5). In this sense

factorised Tate models are the appropriate way to systematically construct the fibrations

that account for the spectrum of possible embeddings of U(1) symmetries arising from

breaking E8 to SU(5). Most surprisingly, however, this structure is in general not respected

by the SU(5) singlet states charged under the U(1). Indeed we will exemplify for the

– 3 –

JHEP03(2013)098

SU(5)×U(1)PQ model that, contrary to expectations based on local reasoning, the entire

spectrum does not assemble into a single E8 representation once we take all singlets into

account. Since the singlets are localised on curves away from the GUT divisor, such

behavior can only be detected in a global approach. However, the singlet curves do intersect

the GUT divisor in points at which couplings to the SU(5) charged matter are localised,

and they are thus also relevant for local model building and phenomenology. To the extent

that the most general pattern of Yukawa couplings cannot, as we find, be unified in a

single E8-point, the idea of a general underlying E8 symmetry present along the entire

GUT divisor cannot be maintained.

Another crucial aspect of model building in F-theory is background gauge flux and

this is intimately related with extra U(1) symmetries as well. Within a global setting our

current understanding of gauge flux requires a fully resolved and smooth manifold and is

realised on the M-theory side as the 4-form G4-flux. For pure SU(5) models G4-flux has

been studied in [44, 56, 57]. This flux was identified in [14] as a massive U(1) flux from a

Type IIB perspective.1 In the presence of massless U(1) symmetries, the associated G4-flux

was studied in [11] for a U(1) model, and in [12–14] for an SU(5)×U(1) model based on the

U(1)-restricted Tate model. Crucial to the definition of the flux is the full resolution of all

the singularities as the expression of G4 involves the resolution divisors from both the SU(5)

singularity and the additional SU(2) singularity. Since one of the primary motivations for

constructing additional U(1) symmetries is that the flux associated to them can induce

chirality in the visible sector, in this paper we also present a construction of the associated

G4-flux.

Our construction fits nicely into the approach of [17], which gives the general form

of the Weierstrass equation for an elliptic fibration with two sections (and therefore one

non-Cartan U(1)), but which is otherwise generic, i.e. has a priori no non-Abelian gauge

symmetries built in. This singular Weierstrass model is resolved in [17] and described by a

smooth fibration with P1,1,2[4] fiber. We show that the factorised Tate models correspond-

ing to SU(5)×U(1)X and SU(5)×U(1)PQ can be mapped to a specialisation (due to the

extra SU(5) symmetry) of the model of [17] which is particularly useful for studying the

U(1) charged singlets. In fact the appearance of a rich pattern of such singlets had been

observed already in [17], albeit in a different context.

The paper is set out as follows. In section 2 we introduce factorised Tate models and

identify the appropriate sections and singularities corresponding to the U(1) symmetries.

We then explicitly resolve the two possible types of SU(5)× U(1) Tate models presenting

the resolved manifold patchwise. In sections 3 and 4 we proceed to analyse in detail

the fibre structure of these SU(5) × U(1)X and SU(5) × U(1)PQ models and derive the

matter spectrum. We work out the fibre structure over a selection of Yukawa points and in

particular present the form over the point corresponding to a 1 10 10 coupling. We also

present the G4-flux associated to the U(1) symmetries constructing the associated two-form

through the Shioda map [58, 59]. In section 5 we study the map between the factorised Tate

models with a single U(1) and the general two-section models of [17]. This in particular

1See the last two references in [45–50] for the analogous fluxes in E6 and SO(10) models, respectively.

– 4 –

JHEP03(2013)098

confirms the presence of a novel type of singlets in the U(1)PQ model which does not fit

into the pattern of a single underlying E8. We summarise our results in section 6. In

appendix A we present the factorised Tate models for the other possible embeddings of

U(1)s in E8 including multiple U(1)s up to the maximum four. In appendix B we explain

the relation between our approach and the U(1)-restricted Tate model as well as the local

split spectral cover.

2 U(1) symmetries from the factorised Tate model

2.1 Engineering extra sections by factorisation

F-theory compatifications to four dimensions are defined in terms of an elliptically fibred

Calabi-Yau 4-fold Y4 : T 2 → B with a section that specifies the base B as a submanifold

of Y4. This universal, so-called zero section is required so as to interpret B as the physical

compactification space. In F-theory massless (up to G4-flux induced Stuckelberg masses)

U(1) symmetries are counted by the number of additional sections [8, 9]. A section specifies

a point in the torus over every point in the base. We write the torus as the Weierstraß

equation

y2 = x3 + fxz4 + gz6 (2.1)

in weighted projective space P[2,3,1] with homogeneous coordinates [x, y, z]. A section is

now specified by two holomorphic equations in [x, y, z] whose intersection lies on the torus.

For the special case of the zero section the two polynomials are z = 0 and the Weierstraß

equation. In this paper we are interested in a specific set of sections that are defined for

cases where the elliptic fibration can be written in the Tate form

y2 = x3 + a1xyz + a2x2z2 + a3yz

3 + a4xz4 + a6z

6 . (2.2)

This defines the Calabi-Yau 4-fold Y4 as a hypersurface in an ambient 5-fold X5. It is

generally not always possible to write the elliptic fibration in this way while retaining the

holomorphicity of the ai, but in the case of SU(5) GUT models it was shown in [40] that

it is possible at least at leading order in the SU(5) divisor, which we denote by2

W : w = 0 . (2.3)

Such Tate models which support an SU(5) singularity on w = 0 are given by the speciali-

sation of the ai to the form [39]

a1 = generic , a2 = a2,1w , a3 = a3,2w2 , a4 = a4,3w

3 , a6 = a6,5w5 . (2.4)

Here the ai,n are functions of the base coordinates, including w, but which do not vanish

at w = 0 so that the assignment (2.4) fixes the vanishing order of the ai at w = 0.

The class of sections we are interested in is, in the Tate form (2.4), defined by the

equation3

y2 = x3 . (2.5)

2Since the U(1)s are global objects we require the fibration to take the Tate form at all orders in w.3Note that in [44, 56, 57] the section (2.5) was termed the Tate divisor and was conjectured to be

the global extension of the spectral cover. We discuss the relation to the local spectral cover more in

appendix B.2.

– 5 –

JHEP03(2013)098

For generic ai,n this defines the zero section z = 0 only. However for special forms of the

ai,n it will define a whole class of sections, and these additional ones correspond to U(1)

symmetries.

To deduce the form of the ai,n it is useful to rewrite (2.5) in terms of the variable

t ≡ y

x(2.6)

as

x = t2. (2.7)

Note that (2.7) implies that in specifying the section an equation in t is holomorphic and

well behaved at x = 0. Now using (2.6) we see that Y4 is given by the vanishing locus of

the Tate polynomial

PT = x2(x− t2) + x2tza1 + x2z2a2,1w + txz3a3,2w2 + xz4a4,3w

3 + z6a6,5w5 (2.8)

inside X5. The section is specified by (2.7) on Y4, i.e. by the vanishing of

X = 0 ∩ PT = 0 (2.9)

inside X5, where we defined

X ≡ t2 − x . (2.10)

Note that

PT |X=0 = t5za1 + t4z2a2,1w + t3z3a3,2w2 + t2z4a4,3w

3 + z6a6,5w5 (2.11)

and for generic polynomials ai,n the only holomorphic solution is at z = 0.

In this form the condition for existence of further sections of the type X = 0 be-

comes obvious, namely PT |X=0 must factorise holomorphically such as to allow for extra

holomorphic zeroes in addition to the universal solution z = 0, i.e.

PT |X=0 = −zn∏

i=1

Yi (2.12)

for some holomorphic polynomials Yi. This in turn implies

PT = XQ− zn∏

i=1

Yi (2.13)

with Q a holomorphic polynomial as well, and therefore Y4 is given by the hypersurface

XQ = zn∏

i=1

Yi ⊂ X5 . (2.14)

Once the polynomials ai,n are restricted in such a way that (2.13) holds, the 4-fold Y4exhibits n obvious sections

X = 0 ∩ Yi = 0, i = 1, . . . , n (2.15)

– 6 –

JHEP03(2013)098

in addition to the zero section at z = 0. The relevance of global factorisations of the spectral

cover equation in heterotic models with extra U(1) symmetries has been investigated in [55].

The sections (2.15) are not all independent because their product generically will in-

clude a term proportional to z5 which is absent from the Tate form (2.2). Therefore there

is one constraint on their coefficients for such a term to be absent. This constraint is

the tracelessness constraint. If we think of the SU(5) as emerging as the commutant of

an SU(5)⊥ inside an underlying E8 as is traditionally done in the context of the local

Higgs bundle picture, the tracelessness constraint ensures that the U(1)s are embedded

into SU(5)⊥ rather than U(5)⊥. In terms of the points on the torus corresponding to the

sections it implies that the sum of them gives back the zero section at the origin [64]. Thus,

an n-fold factorisation as in (2.13) corresponds to n− 1 independent extra sections.

An important aspect of the sections (2.15) is that the Yi are generally not linear

polynomials. This means that the equation (2.15) for a given Yi in fact defines a number of

points corresponding to the multiple roots of the polynomial Yi = 0. We define the section

as the torus sum of these points (see [17, 55] for example on how the addition of points

on the torus fibre is performed). The individual roots themselves still hold information

though as extra matter states localise on loci where two of the roots degenerate. In the

presence of extra non-Abelian gauge symmetry G these states are singlets under G. We

will show that when the two roots come from different Yi factors the associated singlets are

charged under the U(1)s. It is also natural to expect that when the roots are in the same

Yi factor the singlets are neutral under all U(1)s, though such states are more difficult to

identify as they do not correspond to a singularity on the manifold. We will show that it is

also possible to combine these possibilities with two pairs of roots degenerating from each

factor leading to four degenerate roots and in this case doubly charged singlets localise.

We discuss this in more detail in section 4 for a particular example.

To understand how the extra sections give rise to a U(1) symmetry, we note that as

it stands, (2.13) is singular — even away from the obvious A4 singularity in the fiber over

the SU(5) divisor w = 0. This is because the equation is in so-called binomial form, whose

importance in F-theory was stressed more recently in [11, 43]. The singularities of PT not

owed to the SU(5) gauge group arise at the intersection of

X = 0 ∩ Q = 0 ∩ Yi = 0 ∩ Yj = 0 , (2.16)

which for each pair of i, j describes a curve of singularities. Note that because X = 0 is part

of the singularity it is in the patch where the variable t is holomorphic and well defined.

However we should be careful when analysing singularities on the particular locus x = 0

since they depend on a derivative analysis which does not hold generally on this locus:

the manifold could still remain smooth and the apparent singularity due to the binomial

form is misleading. The potential singular nature of the 4-fold at x = 0 must therefore

be checked by going back to the original Weierstrass formulation of the model with fibre

coordinates x, y, z. As we will show this only affects a certain class of SU(5) singlets which

localise on curves that we discuss in section 4.

In particular for the case of a single U(1), and therefore two splitting factors Y1 and

Y2, we have a conifold singularity over the curve X = Q = Y1 = Y2 = 0 [10, 11]. In fact the

– 7 –

JHEP03(2013)098

fiber over this curve exhibits an SU(2) singularity so that we will refer to this singular locus

as the curve of SU(2) singularities. These singularities must be resolved. We will denote

the resolved 4-fold by Y4.4 The resolution introduces new divisor classes Si in H1,1(Y4).

These Si are then related to elements wi ∈ H1,1(Y4) such that expansion of the M-theory

3-form C3 as

C3 = Ai ∧ wi + . . . (2.17)

gives rise to gauge potentials Ai of the Abelian symmetry Ai. We will determine the

relation between Si and wi in detail in section 3.2.

This approach allows for a systematic construction of extra U(1) symmetries for Tate

models by classifying all possible factorisations of the PT |X=0 of the form (2.13). Since

PT |X=0 is a polynomial of degree 5 in t, this amounts to making a general ansatz for the

degree ni polynomials Yi with∑

i ni = 5, subject to extra constraints such that PT |X=0 =

−z∏

i Yi. For example, if we are interested in one extra section, there are two inequivalent

classes of factorisations because the degrees of Y1 and Y2 can be (n1, n2) = (1, 4) or (2, 3).

We will give the explicit form of the factors Y1, Y2 and Q in section 2.3.

Note that a specific global SU(5) Tate model with one extra U(1) was introduced as

the U(1) restricted Tate model in [10], and [11] showed that this model can be brought in

the form (2.13) with n = 2. We elaborate further on the relation of the factorised Tate

models to the U(1) restricted Tate model approach in appendix B.1.

The class of global Tate models has a well-defined local limit w → 0, in which it flows

to the so-called spectral cover or Higgs bundle construction of local models [18]. We review

this limit in appendix B.2. Correspondingly, our factorised Tate models (2.13) precisely

flow to what is called split spectral cover models in the local F-theory literature [18–21]. It

is therefore clear that the constraints on the coefficients ai,n are identical to the constraints

on the sections on W which define the split spectral covers. It is important to stress,

though, that the existence of a U(1) symmetry and the associated U(1) fluxes can never

be determined in a satisfactory manner by focusing only on the local limit. Conceretely

the factorised Tate model constrains also higher order terms in w which do not feature

in the spectral cover limit. The U(1) symmetry is sensitive to the full global details of

the compactification [10, 38]. This in particular requires a full resolution of the binomial

singularities (2.16) to determine the resolved version of the extra sections [10–13]. The

factorised Tate model (2.13) can be viewed as the correct global extension of the local split

spectral cover models.

2.2 Resolving the SU(5) singularity

The discussion just presented was phrased in the limit where Y4 exhibits an SU(5) sin-

gularity in the fiber over w = 0. In order to fully analyse the model, however, we are

interested in understanding the sections after resolving the SU(5) singularity. This resolu-

tion process has been studied with different techniques in the recent F-theory SU(5) GUT

literature in a number of papers [12–14, 41–44] (see [45–50] for other gauge groups) and

4We do not distinguish in notation between the 4-fold where only the SU(2) singularities are resolved or

where also the SU(5) singularity over the divisor W are resolved.

– 8 –

JHEP03(2013)098

we use the process described in [12, 14]. The resolution is achieved through a sequence of

4 blow-ups. This introduces 4 resolution divisors Ei : ei = 0, i = 1, . . . , 4 and amounts to

the replacement

x→ xe1e22e

23e4 , y → ye1e

22e

33e

24 , w → e0e1e2e3e4 . (2.18)

Accordingly the Tate polynomial reads

PT = e21e42e

53e

34 [x3e1e

22e3 − y2e4e3 + a1xyz + a2,1x

2z2e0e1e2 + a3,2yz3e20e1e4

+a4,3xz4e30e

21e2e4 + a6,5z

6e50e31e

24e2] . (2.19)

The proper transform PT is obtained by dividing by the overall factor and describes the

resolved Calabi-Yau 4-fold Y4 as the hypersurface

PT = x3e1e22e3 − y2e4e3 + a1xyz + a2,1x

2z2e0e1e2 + a3,2yz3e20e1e4

+a4,3xz4e30e

21e2e4 + a6,5z

6e50e31e

24e2 (2.20)

inside an ambient 5-fold X5. This ambient space X5 of the resolution is subject to a rich

Stanley-Reisner ideal given by [12, 14]

{xyz, xye0, xe0e3, xe1e3, xe4, ye0e3, ye1, ye2, ze1e4, ze2e4, ze3, e0e2} (2.21)

and one possible choice from the combinations

{ye0

ze4

}⊗

xe0, xe1

xe0, ze2

ze1, ze2

⊗e0e3, e1e3

e0e3, e2e4

e1e4, e2e4

. (2.22)

The different choices correspond to different triangulations. For definiteness we will work

in the sequel with one particular triangulation corresponding to the choice of elements

{xyz, zei|i=1,...,4, xye0, xe0e3, xe1e3, xe4, ye0e3, ye1, ye2, ze1e4, ze2e4, e0e2, e4e1, e4e2} .(2.23)

Note, however, that the specific form of the resolved fiber may dependent on the concrete

triangulation under consideration.

We now wish to apply the same logic as in section 2.1 to describe U(1)s after the SU(5)

resolution. The first thing to specify is the class of sections analogous to (2.5). We take

this to be

y2e4 = x3e1e22 (2.24)

in view of the quantities appearing in (2.20) after dividing by a factor of e3. We define t

as in (2.6) but with x and y the coordinates appearing in (2.20). Note that because the

coordinate transformation (2.18) acting on t is holomorphic in the ei the potential subtlety

discussed in the previous section remains only on the locus x = 0. Suppose the Tate

model prior to SU(5) resolution takes the factorised form (2.13). Since all we have done to

– 9 –

JHEP03(2013)098

Factorisation pattern Number of U(1)s

Y(1)1 Y

(4)2 1

Y(2)1 Y

(3)2 1

Y(1)1 Y

(1)2 Y

(3)1 2

Y(1)1 Y

(2)2 Y

(2)3 2

Y(1)1 Y

(1)2 Y

(1)3 Y

(2)4 3

Y(1)1 Y

(1)2 Y

(1)3 Y

(1)4 Y

(1)5 4

Table 1. Possible factorisation patterns of PT |X=0. The superscripts denote the degree in t.

arrive at (2.19) is to transform coordinates as (2.18), the resulting PT as given in (2.19) is

guaranteed to factorise on the locus X = 0, where now

X = t2e4 − xe1e22 . (2.25)

However, what is not guaranteed is that the proper transform PT given in (2.20) also

factorises into holomorphic components since we have divided out by the prefactor in (2.19).

Indeed it does not. This can be checked on a case-by-case basis as demonstrated in the

following sections. However the meromorphicity arises purely from the resolution divisors

e1 and e2. With the choice of triangulation (2.23) it is simple to check that these divisors

do not intersect the section and therefore the singularity because the Stanley-Reisner ideals

forbids the intersection of (2.24) with e1 = 0 and e2 = 0. In fact also e4 does not intersect

the section and so only e0 and e3 are relevant.

The result that e1, e2 and e4 do not intersect the section implies that in order to resolve

the binomial singularity (2.16) we can work in a patch where we set e1 = e2 = e4 = 1. In

this patch the resolved Tate form does split holomorphically over the section (2.24) and

can be again written as (2.14) with holomorphic Q and the Yi. Therefore, in this patch,

we can resolve the additional singularity and account for the U(1)s. This will be worked

out for the individual factorisations in the next section.

Let us stress that in section 5 we will provide a rather different resolution of the

factorised Tate models based on a P1,1,2[4]-fibration that had appeared before in [17]. In

this approach we will not need to work patchwise, which is more gratifying from a formal

perspective. However, since the actual structure of the factorised Tate models is more

evident and intuitive in the current framework we find it useful to present the analysis of

the matter spectrum etc. in this fashion in sections 3 and 4.

2.3 Factorised Tate models

Having outlined the general approach and formalism we can tackle the specific factorisations

as given in table 1. In this section we work out the explicit form of the equations for the

4− 1 and the 3− 2 factorisations. The other cases are presented in appendix A.

– 10 –

JHEP03(2013)098

2.3.1 4 − 1 factorisation

The 4−1 factorisation corresponds to writing PT |X=0 = −zY1Y2 with Y1 and Y2 polynomials

of respective degrees 1 and 4 in t. Performing the resolution (2.18) on the general factorised

form gives, after the proper transform,

Y1 = c1t+ c0e0z, Y2 = e24(t4d4 + t3e0zd3 + t2e20z

2d2 + d1te30z

3 + d0e40z

4). (2.26)

Here we have set e1 = e2 = 1 because, as discussed in the previous section, these two

resolution divisors do never intersect the extra section. Comparing the above ansatz with

PT |X=0 reveals that the polynomials ci and di are subject to the tracelessness constraint

c1d0 + c0d1 = 0 (2.27)

because there is no term of order t in PT |X=0. As discussed this is a consequence of the

fact that the Tate model has no z5 term.

As mentioned before and discussed in greater detail in appendix B.2, the factorised

Tate model asymptotes, in the local limit w → 0, to the split spectral cover models.

Indeed the factorisation structure and in particular the constraint (2.27) are as for the

U(1)X spectral cover worked out in [19]. The solution to this constraint can be written as

d0 = αc0 , d1 = −αc1 (2.28)

with α some polynomial on B of appropriate degree. Note that we must impose that c0and c1 should not vanish simultaneously in order not to induce non-Kodaira singular fibres

because at this locus all the an,i vanish.

Given Y1 and Y2 we can now explicitly evaluate also the polynomial Q and arrive at

the following parametrisation of the 4− 1 factorised Tate model,

Y1 = tc1 + c0u ,

Y2 = e24(t4d4 + t3ud3 + t2u2d2 − αc1tu3 + αc0u

4)

X = t2e4 − x , (2.29)

Q = e3x2 + c1d4e4t

3z + c1d3e4t2uz + c0d4e4t

2uz + c1d2e4tu2z + c0d3e4tu

2z − αe4c21u3z+c0d2e4u

3z + c1d4txz + c1d3uxz + c0d4uxz

with u = e0z. The case where the SU(5) is unresolved is reached simply by setting e3 =

e4 = 1 and e0 = w. Finally let us briefly describe the resolution of the binomial singularity

X = Q = Y1 = Y2 = 0. We stress again that this singularity lies entirely in the patch

e1 = e2 = 1. Such type of binomial singularities has been introduced recently in [43] in

the context of SU(5) models without extra U(1)s and in [11], which has brought the U(1)

restricted Tate model of [10] into binomial form. The small resolution proceeds by replacing

the singularity in the fiber over the curve X = Q = Y1 = Y2 = 0 by a P1 parametrised by

homogeneous coordinates [λ1, λ2]. This is achieved by describing Y4 in the given patch as

the complete intersection

Y1λ2 = Qλ1 ∩ Y2λ1 = Xλ2 (2.30)

– 11 –

JHEP03(2013)098

inside a 6-fold X6. Away from X = Q = Y1 = Y2 = 0, the extra section is given by the

locus

λ1 = 0 ∩ X = 0 ∩ Y1 = 0 (2.31)

inside X6 as follows by plugging X = Y1 = 0 into (2.30). At X = Q = Y1 = Y2 = 0,

on the other hand, [λ1, λ2] are unconstrainted and therefore the section wraps the entire

resolution P1 as in [10, 11]. This behaviour will be discussed in greater detail in section 3.

2.3.2 3 − 2 factorisation

The 3− 2 factorised Tate model is based on the ansatz

Y1 = c2t2 + c1te0z + c0e

20z

2, Y2 = e24(d3t3 + d2e0t

2z + d1tz2 + d0e

30z

3) (2.32)

subject to the constraint

c1d0 + c0d1 (2.33)

from a5 = 0 in the Tate polynomial. As in the local split spectral cover version a way to

solve the tracelessness constraint is to write [22]5

c0 = αβ , c1 = αδ , d0 = γβ , d1 = −γδ (2.34)

with α, β, γ and δ arbitrary polynomials of appropriate degrees. In order to forbid non-

Kodaira singularities one must impose that the following intersections should be empty,

c2 · α , c2 · β · δ , d2 · d3 · γ . (2.35)

It is worth noting that the constraints will expand if some of the facctors are set to zero

identically over the full 4-fold.

With this information one can again compute Q and arrive, in the patch e1 = e2 = 1,

at the binomial form

Y1 = c2t2 + αδe0tz + αβe20z

2 ,

Y2 = e24(d3t3 + d2e0t

2z − δe20γtz2 + βe30γz3) ,

X = t2e4 − x , (2.36)

Q = e3x2 + c2d3e4t

3z + c2d3txz + c2d2e0e4t2z2 + αd3δe0e4t

2z2 + c2d2e0xz2

+αd3δe0xz2 + αβd3e

20e4tz

3 + αd2δe20e4tz

3 − c2δe20e4γtz3 + αβd2e30e4z

4

+βc2e30e4γz

4 − αδ2e30e4γz4 .5Note that it seems we are writing 4 parameters in terms of 4 other parameters while solving a constraint

a5 = 0, which is not possible. Indeed there are only 3 independent parameters in the ansatz (2.34) since

one can write c1 = c0(δβ

)and d1 = d0

(δβ

). However the important point noted in [22] is that taking

the solution (2.34) allows for additional freedom in distributing the possible globally trivial components of

the matter curves and so can be important when considering the restriction of hypercharge flux to matter

curves.

– 12 –

JHEP03(2013)098

3 Fibre structure and charges in the U(1)X model

We now analyse the fiber structure of our F-theory compactification, starting, in this

section, with the 4 − 1 factorisation. The associated Abelian gauge symmetry is often

referred to as U(1)X in the model building literature. Over generic points on the SU(5)

divisor W : w = 0 in the base B, the fiber can be described in terms of the hypersurface

equation (2.20) within X5. The resolution of the SU(2) singular locus is described by

the complete intersection (2.30) within X6. The Yukawa points on W where SU(5) matter

couples to the singlets localised along the SU(2) curve can also be treated in this approach as

we will see. In total this gives us access to the fiber structure over the entire Calabi-Yau Y4.

3.1 Structure of the matter surfaces

We begin with the fiber structure over W . In analysing the resolution P1s over W and over

the various matter curves we follow the procedure described in [12] (see also [13, 14, 43–50]).

As usual the fiber over generic points on the SU(5) surface W : w = 0 in the base

B is given by a tree of P1s intersecting like the affine Dynkin diagram of SU(5). These

P1i , i = 0, . . . , 4 are the fibers of the resolution divisors Ei : ei = 0 and can be described as

the complete intersection

P1i : ei = 0 ∩ PT = 0 ∩ Da = 0 ∩ Db = 0 ⊂ X5 (3.1)

with Da, Db denoting the pullback of two base divisors that intersect W exactly once. The

intersection of these divisors is such that∫Y4

Ei ∧ Ej ∧Da ∧Db = Cij

∫BW ∧Da ∧Db (3.2)

with Cij the Cartan matrix of SU(5) in conventions where the diagonal has entries −2.

Over the matter curves on W some of these P1s split and assemble into the affine

Dynkin diagram of higher rank groups.

10-matter curves. We first turn to the intersection curve of the Tate polynomial a1 = 0

with W in the base, which, in an SU(5) Tate model, corresponds to the 10 matter curve.

Since in the U(1)X model a1 = c1 d4 there are now two 10 curves

C10(1) : d4 = 0 ∩ w = 0, C10(2) : c1 = 0 ∩ w = 0. (3.3)

As discussed in greater detail in appendix B.2 the structure of the SU(5) charged mat-

ter curves coincides with the corresponding split spectral cover model [19] to which our

construction flows near the SU(5) divisor. Note that C10(2) did not appear in the U(1)

restricted Tate model of [12], where c1 was set to 1, see appendix B.1.

The P1-structure over both 10-curves turns out to be very similar. To describe the

fiber we must specialise, say, Db = 0 in (3.1) to d4 = 0 or c1 = 0, respectively. As a

consequence the polynomial PT will factorise for certain P1i. Such a factorisation indicates

a splitting of P1i s over the matter curve.

– 13 –

JHEP03(2013)098

Concretely over C10(1) this procedure yields the following equations (omitting for

brevity the universal piece d4 = 0 ∩Da = 0) and corresponding P1-splits,

e0 = 0 ∩ e3 (−e1 + e4) = 0 ←→ P10 → P1

03 ∪ P10A,

e1 = 0 ∩ e3 = 0 ←→ P11 → P1

13,

e2 = 0 ∩ e3 − (c1d2 + c0d3) e1 z3 = 0 ←→ P1

2 → P12B1

, (3.4)

e3 = 0 ∩ e0 e1C = 0 ←→ P13 → P1

03 ∪ P113 ∪ P1

3C1,

e4 = 0 ∩ e3 + c1 d3 e0 z2 = 0 ←→ P1

4 → P14D1

,

where we have exploited the Stanley-Reisner ideal to set as many coordinates to one as

possible.6 A factorisation of the above defining equations indicates a splitting of P1s over

the matter curve into the indicated P1s. Note that P113 and P1

03 appear with multiplicity

two. It is now easy to compute the intersection structure of these six P1s by counting

simultaneous solutions to these equations within X5. For instance, since e2e4 is in the

Stanley-Reisner ideal, P14D1

and P12B1

do not intersect. On the other hand, P103∩P1

4D1= 1

because D1|e3=0=e1 vanishes identically so that this intersection is described by the trans-

verse intersection of five polynomials e0 = e3 = e4 = d4 = Da = 0 within X5. In this fashion

one establishes that the six P1s intersect like the nodes of the affine Dynkin diagram of

SO(10) as required in the theory of Kodaira fibers.

Over C10(2) , the exact form the of the defining equations differs slightly, but the P1s

split in an analogous manner into P10A, P1

23, P12B2

, P13C2

, P14B2

, each with multiplicity one,

and P113 and P1

03 each with multiplicity two. The intersection structure is again as in the

affine Dynkin diagram of SO(10).

To identify the combinations of P1s corresponding to the 10 representation one must

compute the Cartan charges of the P1s and compare these to the 10 weights. We observe

that the structure of the matter surfaces is identical to the 10 curve in the U(1) restricted

Tate model as analysed in [12] for the analogous choice of Stanley-Reisner ideal. Therefore

we can refer to [12], section 3.3 for the computation of the SU(5) Cartan charges of the above

P1s and to tables A.18 and A.19 for the resulting identification of suitable combinations

of P1s with the weight vectors of the 10 representation of SU(5). For convenience of the

reader we recall this procedure for the SU(5) Cartan charges of P103 over C10(1) . The gauge

potential associated with the Cartan U(1)i ⊂ SU(5) arises by expanding the M-theory

3-form as C3 = Ai ∧ Ei + . . ., where Ei denotes the 2-form dual to the resolution divisor

ei = 0. Therefore the charge under the generator of U(1)i ⊂ SU(5) is given by the integral∫P103Ei, i = 1, . . . , 4. This can be computed as the intersection

ei = 0 ∩ e0 = 0 ∩ e3 = 0 ∩ d4 = 0 ∩ Da = 0 ⊂ X5. (3.5)

For i = 1 and i = 4 this is the transverse intersection of five degree-one polynomials

inside X5, which have one intersection point. For i = 2, on the other hand, this vanishes

because e0e2 is in the Stanley-Reisner ideal. Finally for i = 3 we do not encounter an

6The polynomial C1 takes the form C1 = c1e2x(d1e20e1e4 + d3x) + c1d2e0e4y+ c0e0e4(e0e1e2(d0e

20e1e4 +

d2x) + d3y).

– 14 –

JHEP03(2013)098

effective intersection must therefore use the following trick: we first note that the integral∫P10AE3 = 1, and that the generic intersection of e3 = 0 with P1

0 over a generic point on

the SU(5) divisor W : w = 0 in the base vanishes because of (3.2). Since P10 splits into P1

03

and P10A, this implies

∫P103E3 = −1. Therefore the U(1)i charges of P1

03 are [1, 0,−1, 1],

corresponding to the weight µ10 − α2 − α3 of the 10 representation of SU(5).

The Cartan charges over the second 10 curve work out in exactly the same man-

ner. For convenience the P1-combination for the various states of the 10-representation is

summarised in the following table (valid for both 10 matter curves):

Weight P1i − combination

µ10 2P103 + P1

0A + P113 + P1

4Di

µ10 − α2 2P103 + P1

0A + P113 + P1

2Bi+ P1

4Di

µ10 − α1 − α2 2P103 + P1

0A + 2P113 + P1

2Bi+ +P1

4Di

µ10 − α2 − α3 P103

µ10 − α1 − α2 − α3 P103 + P1

13

µ10 − α2 − α3 − α4 P103 + P1

4Di

µ10 − α1 − 2α2 − α3 P103 + P1

13 + P12Bi

µ10 − α1 − α2 − α3 − α4 P103 + P1

13 + P14Di

µ10 − α1 − 2α2 − α3 − α4 P103 + P1

13 + P12Bi

+ P14Di

µ10 − α1 − 2α2 − 2α3 − α4 2P103 + P1

2Bi+ P1

3Ci+ P1

4Di

(3.6)

To see the difference between both 10 curves we must investigate the intersection

pattern of the P1s with the extra section S. Since the resolution divisors e1, e2 and e4 do

not intersect the section, the only possible intersections occur for P103 as well as for P1

3C1

and P13C2

(over d4 = 0 or c1 = 0, respectively). Therefore it is sufficient to carry out the

analysis of the intersection pattern with S inside the complete intersection X6. Recall that

at generic points away from the SU(2) singularities the resolved section S is given by the

locus

λ1 = 0 ∩ X = 0 ∩ Y1 = 0 (3.7)

inside the complete intersection (2.30). The intersection number between one of the above

P1s and the extra section S is counted by the number of generic simultaneous solutions of

the defining equations within X6. A priori these are seven constraints within the ambient

6-fold and thus have no common solution. However, it can happen that not all of these are

independent. If we end up with precisely 6 mutually non-exclusive independent constaints,

the intersection number is non-zero.

To simplify the expressions we will set e1 = 1, e2 = 1, e4 = 1 as these are non-vanishing

in the complete intersection patch and also z = 1 because ze3 is in the Stanley-Reisner

ideal. We start with the fiber over C10(1) corresponding to d4 = 0. Along P103, the two

constraints Y1 = 0 and X = 0 appearing in S evaluate to

c1t = 0 ∩ t2 − x = 0. (3.8)

– 15 –

JHEP03(2013)098

The only solution over generic points on d4 = 0 is t = 0 = x and thus x = y = 0, but xye0is in the Stanley-Reisner ideal. Thus P1

03 does not intersect the section in the fiber over

d4 = 0.

Concerning the intersection of P13C1

with the section we note that the constraint C1 = 0

is automatically fulfilled once we set e3 = 0 and Y1 = X = 0. Therefore we end up with

the six independent constraints

x= t2 ∩ c1t+c0e0=0 ∩ e3=0 ∩ Da=0 ∩ λ1=0 ∩ d4 = 0. (3.9)

It is important to note that in the present case Y1 = c1t + c0e0 is of degree one in t.

Therefore this system of equations has precisely one solution and thus S ∩ P13C1

= 1.

Over the second 10 matter curve C10(2) , corresponding to c1 = 0, the situation is

reversed: for e0 = 0 = e3 = c1, Y1 vanishes automatically. Thus the intersection of P103

with S is given by the single generic intersection of the six polynomials

e0 = 0 ∩ e3 = 0 ∩ c1 = 0 ∩ Da = 0 ∩ λ1 = 0 ∩ x = t2 (3.10)

within the ambient 6-fold X6. By contrast, now P13C2

has no generic intersection with S

because on this locus Y1 evaluates to c0e0, and the intersection with e0 = 0 had been

accounted for already in P103.

To conclude, the difference between the two 10 curves is the following intersection

pattern with the extra section S:

C10(1) : S ∩ P103 = 0, S ∩ P1

13 = 0, S ∩ P13C1

= 1,

C10(2) : S ∩ P103 = 1, S ∩ P1

13 = 0, S ∩ P13C2

= 0. (3.11)

This difference will be crucial when it comes to computing the U(1)X charges of the states.

5-matter curves. A similar analysis is easily carried out for the 5 matter curves. For

the 4−1 factorisation, the 5 curve P = 0∩w = 0 in the base B splits in the following way:

P =a21a6,5−a1a3,2a4,3+a2,1a23,2 → (d23c0+d2d3c1−d1d4c1)(d4c20+d3c0c1+d2c

21)=:P1P2 ,

(3.12)

where we used the tracelessness constraint d1c0 +c1d0 = 0. Like in the U(1)-restricted case

of [12], over both 5-curves

C5(1) : P1 = 0 ∩ w = 0, C5(2) : P2 = 0 ∩ w = 0 (3.13)

one observes a splitting of P13 into two P1’s,

P13 →

{P13G1∪ P1

3H1for P1 = 0,

P13G2∪ P1

3H2for P2 = 0,

(3.14)

where H1, H2, G1 and G2 are some longish polynomials which we refrain from displaying

here. The remaining P10,P1

1,P14 are unaffected. The intersection pattern of the fibred P1’s is

– 16 –

JHEP03(2013)098

as in equ. (A.31) of [12] and corresponds to the affice Dynkin diagram of SU(6). Moreover,

one readily evaluates the Cartan charges of, say, the splitting P13,

P13Gi : [0, 1,−1, 0] = −µ5 + α1 + α2, P1

3Hi = [0, 0,−1, 1] = µ5 − α1 − α2 − α3.(3.15)

The full identification of all weights in the 5 representation is given as follows:

Weight P1i − combination

µ5 P10 + P1

3Hi+ P1

4

µ5 − α1 P10 + P1

1 + P13Hi

+ P14

µ5 − α1 − α2 P10 + P1

1 + P12 + P1

3Hi+ P1

4

µ5 − α1 − α2 − α3 P13Hi

µ5 − α1 − α2 − α3 − α4 P13Hi

+ P14

(3.16)

While the fiber structure over both 5 curves is identical, the distinguishing property

is again the intersection pattern with the section. Analogous considerations as for the 10

representations yield

C5(1) S ∩ P13G1

= 0, S ∩ P13H1

= 1, (3.17)

C5(2) S ∩ P13G1

= 1, S ∩ P13H2

= 0. (3.18)

SU(5)-singlet curves. There is one more type of matter curves inhabited by U(1)Xcharged singlets. These extra states arise from M2-branes wrapping suitable components

of the fiber over the self-intersection of the I1-part of the discriminant locus. As in [10], the

fiber over this self-intersection locus acquires an SU(2)-singularity prior to resolution. This

curve of SU(2) singularities is a consequence of the binomial structure of the factorised Tate

mode and occurs, before the small resolution, at X = 0 ∩ Q = 0 ∩ Y1 = 0 ∩ Y2 = 0 ⊂ X5.

This describes a curve C1 in the base space times a point (x, t) = (x0, t0) in the fiber

at which the fiber degenerates. After the small resolution (2.30), this singular point is

replaced by the P1 parametrised by the homogeneous coordinates [λ1, λ2], called P1SU(2) in

the sequel. The original fiber, called P10, is the locus X = 0 ∩ Q = 0 ∩ Y1 = 0 ∩ Y2 = 0

away from the point (x0, t0), which has been blown up into P1SU(2). It is therefore clear

that these two fiber components intersect at two points, thus forming the affine Dynkin

diagram of SU(2). This same structure had been discussed before in [10–13, 17].

Note that for the U(1)X model under consideration the curve C1 over which this fiber

is localised is a single connected curve in the base. This follows by explicitly solving for

X = Q = Y1 = Y2 = 0 taking into account both the Stanley-Reisner-ideal and the extra

restriction that c1 = 0 and c0 = 0 are not allowed to intersect. Since this curve does not lie

on top of the SU(5) brane its structure cannot be accounted for in any local model. This

is part of the reason why a global understanding is required in the study of Abelian gauge

groups. M2-branes wrapping P1SU(2) thus give rise to massless SU(5) singlets 1.

The section S is given, away from the critical locus X = 0 ∩ Q = 0 ∩ Y1 = 0 ∩ Y2 = 0

by (2.31). As X = Q = Y1 = Y2 = 0, however, it wraps the entire P1SU(2) because the

homogeneous coordinates [λ1, λ2] are now unconstrained. Therefore S ∩ P10 = 2 and so

C1 : S ∩ P1SU(2) = −1 (3.19)

because S intersects the entire fiber class in a single point, S ∩ (P10 + P1

SU(2)) = 1.

– 17 –

JHEP03(2013)098

3.2 U(1)X charges for matter curves and G4-flux

We now investigate the detailed relationship between the appearance of the extra section

and the appearance of a U(1)X . The relevance of S lies in the fact that its dual 2-form

is related to an element wX of H1,1(Y4) in terms of which the M-theory 3-form C3 can be

expanded as C3 = AX∧wX+. . .. The 1-form AX is then the gauge potential associated with

an extra U(1)X gauge symmetry, a priori in the 3-dimensional effective theory obtained

by dimensional reduction of M-theory on Y4. See [16, 62, 63] for recent investigations of

various aspects of this effective action. To find the precise relation between the dual 2-form

S7 and wX one requires that wX satisfy the relations∫Y4

wX ∧Da ∧Db ∧Dc = 0,

∫Y4

wX ∧ Z ∧Da ∧Db = 0, (3.20)∫Y4

wX ∧ Ei ∧Da ∧Db = 0, i = 1, . . . , 4 (3.21)

with Da, Db, Dc the pullback of arbitrary base divisors. The first two constraints ensure

that under F/M-theory duality AX actually lifts to a 1-form in four dimensions. The last

constraint normalises the U(1)X generator to be orthogonal to the Cartan generators of the

non-Abelian gauge group, in our case SU(5). The first two constraints were worked out for

the U(1) restricted Tate model in [10] (see also [11]) and for the SU(5)×U(1)X restricted

Tate model with a single 10-matter curve in [12] (see also [13]). In the mathematics

literature the map from S to wX is known as the Shioda map [58, 59], as reviewed recently

e.g. in [16, 17].

As for the first constraint, observe that∫Y4

S ∧Da ∧Db ∧Dc =

∫BDa ∧Db ∧Dc (3.22)

because S is a section. Thus we subtract Z because∫Y4

(S − Z) ∧Da ∧Db ∧Dc = 0. Next

we compute∫Y4

(S − Z) ∧ Z ∧Da ∧Db =

∫Y4

S ∧ Z ∧Da ∧Db +

∫BK ∧Da ∧Db, (3.23)

where we used that ZZ = −ZK in terms of the anti-canonical class K of B. The intersection∫Y4S ∧ Z ∧Da ∧Db is evaluated in the complete intersection X6 as the number of generic

intersections of

x = t2 ∩ c1t = 0 ∩ λ1 = 0 ∩ z = 0 ∩ Da = 0 ∩ Db = 0 (3.24)

in X6. The first two constraints are Y1 = 0 and X = 0 evaluated for z = 0. Since

x = t = z = 0 is excluded by the Stanley-Reisner ideal, the intersection is∫Y4

S ∧ Z ∧Da ∧Db =

∫Y4

c1 ∧ Z ∧Da ∧Db =

∫Bc1 ∧Da ∧Db. (3.25)

7Our notation does not distinguish between a divisor D and its dual 2-forms; also, the 2-form dual to

divisors of the form, say, c1 = 0 will be denoted by c1.

– 18 –

JHEP03(2013)098

Thus the first two constraints are satisfied by8 S − Z − K − c1.Finally we implement

∫Y4

wX ∧ Ei ∧Da ∧Db = 0. Since zei is in the Stanley-Reisner

ideal the only constraint arises from the intersection with S, which is given by∫Y4

S ∧ Ej ∧Da ∧Db = δj3

∫BW ∧Da ∧Db. (3.26)

To eliminate this intersection with E3 without spoiling the first two constraints we add a

linear combination∑4

i=1 tiEi such that∑

i ti∫Y4Ei∧Ej ∧Da∧Db =

∑i tiCij

∫BW ∧Da∧

Db = −δj3 with Cij the SU(5) Cartan matrix, cf. eq. (3.2). In total the correct U(1)Xgenerator is

wX = 5(S − Z − K − c1) +∑i

tiEi, ti = (2, 4, 6, 3). (3.27)

Here we have picked the overall normalisation of wX such that no factional charges will

appear. Note that for c1 ≡ 1 this reduces to the expression found in [12] for the SU(5) ×U(1)X restricted Tate model with one 10-curve.

We are now in a position to compute the U(1)X charges of the 10 representation

localised on the the matter curves. These are given by

qX =

∫∑

P1ij

wX (3.28)

with∑

P1ij denoting the linear combination of P1s in the fiber of the respective matter

curves corresponding to one component of the weight vector of the representation. Of

course the value of the integral is the same for all weights. This expression is easiest

computed for the weight µ10 − α2 − α3 corresponding to P103. The integral

∫P103Ei gives

just the Cartan charges [1, 0,−1, 1] of this weight and thus∫P103

∑i tiEi = −1. Furthermore

ze3 is in the Stanley-Reisner ideal so that∫P103Z = 0, as is

∫P103−K − c1.9 Now it becomes

crucial that the intersection pattern of P103 with S differs for the fiber over the two 10

curves as given in (3.11). Adding up these contributions yields

C10(1) : q101 = −1, C10(2) : q10(2) = 4. (3.29)

A similar computation for the 5-curves leads to the charges

C5(1) : q5(1) = 2, C5(2) : q5(2) = −3. (3.30)

8We hope the reader is not confused by the fact that c1 denotes the Tate polynomial in a1 = d4c1 and

not, as oftentimes in the literature, c1(B). We will always express c1(B) in terms of K.9Consider first

∫P103K: for the fiber over d4 = 0 this intersection is e0 = 0 ∩ e3 = 0 ∩ d4 = 0 ∩ Da =

0∩K = 0 ⊂ X5 for an arbitrary divisor in the base Da that intersects the matter curve once. This vanishes

because on the 4-fold, e0 = 0 is constrained to lie over the SU(5) divisor w = 0 in the base, and the generic

intersection of this with the three more base divisors Da = 0, d4 = 0 and K vanishes. The same holds for∫P103K over c1 = 0. By a similar argument

∫P103

c1 = 0 over d4 = 0. Over c1 = 0 on the other hand,∫P103

c1

boils down to∫Y4E0 ∧E3 ∧ c1 ∧Da = C03

∫BW ∧Da ∧ c1 = 0 with C03 = 0 the corresponding entry from

the extended Cartan matrix of SU(5).

– 19 –

JHEP03(2013)098

Finally, the singlets from M2-branes wrapping P1SU(2) have charge −5 as a consequence

of (3.19) and the fact that Z and Ei have zero intersection with P1SU(2),

C1 : q1 = −5. (3.31)

The conjugate singlets are due to M2-branes wrapping P10. This is the component of the

singular SU(2) fiber intersected by the universal section, Z ∩ P10 = 1 so that

∫P10

wX = 5

because S ∩ P10 = 2.

To conclude this section we stress that as in [10–13] the extra U(1)X gauge group

opens up the possibility of switching on associated non-trivial gauge flux. By F/M-theory

duality, such gauge flux is described in terms of the M-theory 4-form field strength

GX4 = F ∧ wX , F ∈ H1,1(B). (3.32)

In particular this induces a chiral spectrum of charged matter states Ri with chiral index

given by ∫CRi

GX4 = qi

∫CRi

F, (3.33)

where the states Ri are localised on the matter surface CRi , which is P1-fibered over the

curve CRi in the base B. With the help of our results for the charges qi the computation

of the chiral index thus reduces to evaluating the integral of the flux F over the matter

curve in the base.

3.3 Yukawa points

We now come to the points of Yukawa interactions at the intersection of the various matter

curves.

An interesting feature compared to the previously analysed U(1)-restricted Tate model

with only a single 10-curve is that the intersection of the two 10 distinct matter curves gives

rise to the Yukawa coupling 10(1)−1 10

(2)−415. This field theoretic expectation is confirmed by

an explicit analysis of the fiber structure over the intersection of C10(1) and C10(2) along

the SU(5) divisor in the base, corresponding to c1 = d4 = w = 0. To this end we can start

from the hypersurface equations (3.4) and set c1 = 0. In particular, P13C1

splits as

P 13C1→ P1

03 ∪ P143 ∪ P1

3C(3.34)

with the polynomial C = (d0e20e1e4 + d2x) + d3y. This leaves us with six P1s (including

multiplicities),

1× P13C, 1× P1

2B, 1× P1

0A, 2× P134, 2× P1

13, 3× P103 (3.35)

with B = e3 − c0d3e1z3. The intersection structure of these P1s follows again by count-

ing common solutions of the involved hypersurfaces with the help of the Stanley-Reisner

ideal (2.22). For example, the intersection P134 ∩ P1

0A is characterised by the six equations

e3 = 0 ∩ e4 = 0 ∩ e0 ∩ e1 = e4 ∩ d4 = 0 ∩ c1 = 0 (3.36)

– 20 –

JHEP03(2013)098

2B 13 03

0A

34 3C

1010 1

SU(2)

Figure 1. Schematic drawing of the 10(1) 10(2)

1 Yukawa coupling in the case of the 4-1 split.

inside the ambient 5-fold X5, which generically has no solution. In other cases some of the

constraints coincide and common solutions are possible. This way one can establish that

the six P1s intersect with one another like the (non-extended) Dynkin diagram of E6.

So far we have not taken into account the SU(2) singularity and its resolution. Recall

from (3.11) that over generic points on the 10 matter curves C10(1) and C10(2) , the resolved

section S intersects P13C1

and P103, respectively. The transition between both intersections

occurs as the two matter curves intersect, where P13C1

splits off an extra copy of P103.

Indeed, at the intersection of the locus c1 = 0 ∩ d4 = 0 ∩ e0 = 0 ∩ e3 = 0 with x = t2, the

polynomials Y1, Y2, X,Q all vanish, indicating that this point lies on the SU(2) singular

locus which is resolved by the small resolution (2.30). The singular point is replaced by

the P1SU(2) described by the homogeneous coordinates [λ1, λ2]. Therefore the central P1

03 of

the E6 Dynkin diagram discussed above is intersected by this extra P1SU(2). The topology

of the fibre including this incoming P1SU(2) is depicted in figure 1. Note that the specific

intersection pattern may depend on the concrete choice of the Stanley-Reisner ideal, i.e.

on the particular triangulation one is working with.

For the intersection of the singlet with the two 5-curves, i.e. the 5(1)−25

(2)−315 Yukawa

coupling, we obtain the same SU(7) pattern as in [12]. Here, as in [12], the P1 corresponding

to the resolution of the SU(2)-singularity ‘appears’ between P13G1

/P13G1

and P13H1

/P13H2

.

The same methods also allow for an analogous analysis of the remaining familiar Yukawa

coupling points between SU(5) charged matter states.

4 Fibre structure and charges in the U(1)PQ model

We now address the 3 − 2 factorised Tate mode of section 2.3.2 in more detail. The

additional U(1) is referred to as of Peccei-Quinn type and denoted by U(1)PQ in the local

model building literature. This corresponds to the fact that for a local 3 − 2 split it is

possible to assign the Higgs up and down multiplets to different matter curves [20, 21] and

so they have different charges under U(1)PQ.

10-curves. Let us begin with the two 10-matter curves

C10(1) : d3 = 0 ∩ w = 0, C10(2) : c2 = 0 ∩ w = 0. (4.1)

An analysis of the P1 split completely analogous to that in section 3.1 yields a fiber structure

over both curves identical to the one given in (3.6), even though, of course, the explicit

– 21 –

JHEP03(2013)098

form of the polynomials B1, C1, D1 and B2, C2, D2 differs. What is very interesting, on

the other hand, are the changes in the intersection structure of P103,P1

13 and P13Ci

with the

section S. This is of course crucial to determine the correct U(1) charges.

The extra section S is still given, over generic loci, by λ1 = 0 ∩X = 0 ∩ Y1 = 0 ⊂ X6,

but with

Y1 = c2t2 + αδe0tz + αβe20z

2, X = t2e4 − x. (4.2)

The big difference to the 4− 1 model is that Y1 is a polynomial of degree 2, not of degree

1, in t. This changes the intersection pattern as follows: first note that the intersection of

S with the resolution P13 over a generic point on the SU(5) divisor w = 0 is (setting z = 1

and e4 = 1)

e3 = 0 ∩ λ1 = 0 ∩ x = t2 ∩ Y1 = 0 ∩ Da = 0 ∩ Db = 0 (4.3)

inside X6 with Da and Db intersecting w = 0 once in the base. Since Y1 is of degree 2, the

intersection number is

P13 ∩ S = 2. (4.4)

Let us now compute the intersections with the descendants of P13 over the 10 curves,

beginning with C10(2) . Concerning P103 we observe that Y1|c2=e0=e3=0 = 0 so that, as is the

case for C10(2) in section 3.1, P103∩S = 1. Unlike before, however, also P1

3C2∩S = 1 because

Y1|c2=e3=0 = αδe0t+αβe20 (after setting z = 1) and if we solve this degree 1 polynomial for

t and plug the solution into C2, the latter vanishes automatically along P13C2

. Note that

indeed P103 ∩ S + P1

3C2∩ S = P1

3 ∩ S.

Over C10(1) , corresponding to d3 = 0, P103∩S = 0 as no simplifications in Y1 occur. To

compute the intersection P13C1

we note that solving Y1|e3=0 = 0 for t gives us two solutions

because Y1 is degree 2. For each of these, C1 vanishes automatically once we impose all

other constraints of the defining equation of P13C1

. Thus P13C1∩ S = 2. To summarise,

C10(1) : S ∩ P103 = 0, S ∩ P1

13 = 0, S ∩ P13C1

= 2,

C10(2) : S ∩ P103 = 1, S ∩ P1

13 = 0, S ∩ P13C2

= 1. (4.5)

5-curves. There are now three 5 curves located at the intersection of the SU(5) divisor

with the zero locus of the three polynomials P1, P2, P3 into which P = a21a6,5−a1a3,2a4,3 +

a2,1a23,2 factorises,

P1 = δ, P2 = βd3 + d2δ, (4.6)

P3 = α2c2d22 + α3βd23 + α3d2d3δ − 2αc22d2γ − α2c2d3δγ + c32γ

2. (4.7)

In the fiber over each of these three matter curves

C5(i) : Pi = 0 ∩ w = 0 ⊂ B, i = 1, 2, 3 (4.8)

one observes that P13 splits according to

P13 → P1

3Gi ∪ P13Hi . (4.9)

– 22 –

JHEP03(2013)098

The explicit form of Gi and Hi is rather lengthy, in particular for i = 3. However, in all

three cases one can easily evaluate the Cartan charges of P13Gi

and of P13Hi

as

P13Gi : [0, 1,−1, 0] = −µ5 + α1 + α2, P1

3Hi = [0, 0,−1, 1] = µ5 − α1 − α2 − α3.(4.10)

In fact, in all three cases the complete weight assignments of the 5 representation are

exactly as in (3.16).

What distinguishes the three 5 curves is the intersection pattern of the respective

fibers with the section S and thus the U(1)PQ charges. An explicit analysis of the defining

polynomials of all the P1s reveals

P13G1∩ S = 2, P1

3H1∩ S = 0, (4.11)

P13G2∩ S = 0, P1

3H2∩ S = 2, (4.12)

P13G3∩ S = 1, P1

3H3∩ S = 1. (4.13)

The logic behind these computations is identical to the 10-curves: concerning P13G1

, one

can solve the quadratic polynomial Y1|e3=0 for t and confirm that for both solutions in t the

polynomial G1 vanishes identically if we take into account all further polynomials entering

the section S and P13G1

. By contrast for P13H1

no such simplifications occur and thus the

intersection number vanishes. On the other hand, for P13G3

it is simpler to solve G3 for x

and combine this with x = t2 into an equation for t, again with two solutions. Crucially,

only one of these solves Y1 = 0, leading to P13G3∩ S = 1 = P1

3H3∩ S.

Singlet curves. The singlet curves in the U(1)PQ model are particularly interesting and

exhibit additional structure. In fact we encounter 3 types of singlets and to see how these

types are classified it is worth discussing in more detail the loci on which singlets are

expected to localise. Consider the section

X = Y1 = 0 . (4.14)

Because Y1 is a quadratic polynomial this defines two points on the torus which are the

two roots of Y1 = 0,

t =e0z

2c2

(−αδ ±

√−4αβc2 + α2δ2

). (4.15)

The first type of singlets are the usual ones as in the U(1)X case. These correspond

to loci where a single root of Y1 coincides with a single root of Y2. Recall that prior

to resolution, the Tate model is singular along the curve X = Q = Y1 = Y2 = 0, and

these singlets localise on the locus C1(1) corresponding to the generic solution of these four

polynomials. As will become clear momentarily, this is the locus away from (x, t) = (0, 0),

C1(1) : X = 0 ∩ Q = 0 ∩ Y1 = 0 ∩ Y2 = 0 , (x, t) 6= (0, 0). (4.16)

After resolution the section S wraps the resolution P1SU(2) in the fiber over C1(1) and by

the same arguments as in the U(1)X model the intersection number is

C1(1) : P1SU(2) ∩ S = −1. (4.17)

– 23 –

JHEP03(2013)098

This leads to singlets with charge ±5 which localise on generic solutions to (4.16).

The second type of singlets are again charged ones that localise on special sub-loci of

the curve (4.16) where additionally two roots in the Y1 and two roots in the Y2 factors

degenerate so that in total four roots degenerate. The loci where this occurs are

C1(2) : x = 0 ∩ t = 0 ∩ β = 0 ∩ δ = 0 , (4.18)

C1(3) : x = 0 ∩ t = 0 ∩ γ = 0 ∩ α = 0 . (4.19)

As discussed in section 2, because these loci are at x = 0 we should check that the expected

SU(2) singularity is present also in the coordinates {x, y}, which indeed can be confirmed.

Already from the previous discussion we can guess the charges of these states: because

charge ±5 singlets are localised where one root from Y1 coincides with one root from Y2,

here we expect states of the double charge, i.e. with q = ±10. Due to the different form of

the binomial singularity on this locus, and potential subtleties on the locus x = 0, we will

not perform the resolution on these special loci in the current approach. Instead we will

explicitly show how to recover the doubly charged singlets associated to C1(2) in a different

formalism in section 5.10

The curve (4.16) also has other special solutions analogous to (4.18)–(4.19) but where

only three roots, rather than four, degenerate. For example the loci x = t = α = β = 0

and x = t = γ = β = 0. However it can be checked that the manifold is not singular on

these loci by directly analysing the Tate polynomial (2.2) in the coordinates {x, y}. More

precisely the singular loci are on the locus a6,5 = a4,3 = a3,2 = 0 where

a6,5 = αβ2γ ,

a4,3 = αβd2 + βc2γ − αδ2γ ,a3,2 = αβd3 + αd2δ − c2δγ . (4.20)

The third type of singlets are associated to loci where a degeneration of roots inside

the same factor Y1 occurs. We expect completely neutral singlets to localise there but since

the manifold is non-singular on this locus this is harder to show explicitly. Note that the

locus where the roots degenerate 4βc2 = αδ2 can be written as

4c0c2 = c21 . (4.21)

The projection of this curve to the GUT brane w = 0 was indeed identified in [20, 21] as

the expected projection of the neutral singlets from the group theory.

U(1)PQ generator and charges. The generator wPQ of the U(1)PQ symmetry is de-

termined by the same logic as in section 3.2. What differs is first that∫Y4S∧Z∧Da∧Db =

10The arguments just given suggest similar results for C1(3) , but our analysis of section 5 is valid only

if γ = 0 ∩ α = 0 is empty so more work is required in cases where this constraint is not met. In practice

we can bypass this problem by restricting ourselves to base spaces B such that α = 0 and γ = 0 do not

intersect, and this is the approach we are going to take from now on.

– 24 –

JHEP03(2013)098

2B 13 03

0A

34

3C

1010 1

SU(2)

Figure 2. Schematic drawing of the 10(1) 10(2)

1 Yukawa coupling in the case of the 3-2 split.

∫B c2∧Da∧Db and second that the intersection number with E3 is now 2, not 1, see (4.4).

This fixes wPQ to take the form

wPQ = 5(S − Z − K − c2) + 2∑i

tiEi, ~t = (2, 4, 6, 3), (4.22)

again for a convenient choice of overall normalization.

Consequently we arrive at the following U(1)PQ charges:

C10(1) : q10(1) = −2, C10(2) : q10(2) = 3, (4.23)

C5(1) : q5(1) = −6, C5(2) : q5(2) = 4, C5(3) : q5(3) = −1, (4.24)

C1(1) : q1(1) = −5. (4.25)

The remaining singlets over (4.18) will be discussed in section 5 and have charges (see

footnote 10)

C1(2) : q1(2) = 10. (4.26)

Yukawa points. The Yukawa coupling 10(1)−210

(2)−31

(1)5 (with the subscripts denoting

U(1)PQ charge for clarity) is located at the triple intersection of C10(1) , C10(2) and the

generic locus (4.16). The same splitting as in (3.35) occurs, the only difference being that

the section S intersects both P103 and P1

3Conce — at least in the current triangulation

used. This modifies the intersection pattern of the fiber as given in figure 2.

The locus of the doubly charged singlets C1(2) intersects two of the 5-matter curves

and the charges precisely agree with an associated cubic interaction

5(1)−6 5

(2)−4 1

(2)10 . (4.27)

It can be checked, in the framework of section 5, that indeed the fibre exhibits the structure

of an SU(7) enhancement at that point.

Implications for group theoretic embedding into E8. The appearance of the extra

charged singlets 1(2)±10 is quite surprising from a group theoretic perspective: the common

lore in the literature is that global Tate models are based on a single E8 gauge group which

is broken to the gauge group G ⊂ E8 along the divisor w = 0. As reviewed in appendix B.2

this breaking can be understood locally in terms of a Higgs bundle [18] with structure

group SU(5)⊥, which in the present case would factorise into SU(3)⊥ × SU(2)⊥ ×U(1)PQ.

– 25 –

JHEP03(2013)098

However, the decomposition of 248 of a single E8 into irreducible representations of SU(5)×SU(3)⊥ × SU(2)⊥ × U(1)PQ only gives rise to a single type of charged singlets 1±5. The

appearance of two types of charged singlets 1(1)±5 and 1

(2)±10 in the factorised U(1)PQ Tate

model implies that no embedding into a single underlying E8 is possible. This conclusion

is also supported by the structure of the Yukawa couplings: a further specialisation of

the complex structure moduli of the 4-fold can lead to points of E8 enhancements on

the divisor w = 0 where the Yukawa couplings 10(2)10(2)5(1), 10(2)5(2)

5(2) and 5(1)5(3)1(1)

come together. This corresponds to the embedding of all these representations into a single

E8. By contrast, the Yukawa coupling 5(1)5(2)1(2) can never coincide with this point of E8

as this would require that c2 = β = δ = 0 on a single point on w = 0, but the intersection

of c2 · β · δ must be forbidden as it would lead to a non-Kodaira singularity, see eq. (2.35).

Put differently, if one were to construct a heterotic dual of a 3 − 2 factorised model

with w = 0 the base of a K3-fibration, the heterotic dual would have to be singular in such

a way as to incorporate the extra charged singlets 1(2)±10 in a non-perturbative fashion as

these cannot arise from the same perturbative heterotic E8 factor as the remaining states.

5 Factorised SU(5) × U(1) models as P[1,1,2]-fibrations

In this section we provide a rather different, but equivalent description of the factorised

Tate models with one extra U(1)-symmetry. Motivated by a study of the landscape of

6-dimensional F-theory compactifications, ref. [17] recently provided the general form of a

Weierstraß equation that decribes an elliptic fibration with two independent sections over

the 2-complex dimensional base space P2. The conclusion of [17] is that such a Weierstraß

model can be written as

Y 2 = X3 +(C1 C2 − B2 C0 − 1

3 C22

)X Z4+

+(C0 C2

3 − 13 C1 C2 C3 + 2

27 C32 − 2

3 B2 C0 C2 + 14B2 C2

1

)Z6 .

(5.1)

Here the fiber coordinates X,Y, Z are homogeneous coordinates on P[2,3,1] and B and Ci

denote some generic sections of some line bundles over the base, which in the case of [17]

was taken to be P2. Indeed, for a Weierstraß model of the form (5.1) one finds that

[X,Y, Z] = [C23 − 2

3 B2 C2, −C33 + B2 C2 C3 − 1

2 B4 C1, B] (5.2)

solves the Weierstraß equation and therefore represents an additional section besides the

universal zero section Z = 0.

For the same reasons as in the factorised Tate models the restriction (5.1) of the Weier-

straß model responsible for this extra section renders the model singular in codimension

2. In [17] these singularities are resolved by translating the P[2,3,1]-fibration into a P[1,1,2]

fibration with homogeneous coordinates w, v, u and then blowing up the point w = u = 0

in the fibre. This introduces a blow-up divisor with coordinate s. The resolved space then

takes the form of a Bl[0,1,0]P[1,1,2]-fibration

B v2 w + sw2 = C3 v3 u + C2 s v2 u2 + C1 s2 v u3 + C0 s

3 u4 , (5.3)

– 26 –

JHEP03(2013)098

with w, v, u and s the homogeneous coordinates of Bl[0,1,0]P[1,1,2]. Indeed this blow-up

procedure for the fiber over the SU(2) singular curve and the resulting transition to a

Bl[0,1,0]P[1,1,2] fibration had also been applied in the U(1) restricted Tate model [17], which

is a special case of the model (5.3). See furthermore [52–54, 65, 66] for the relevance of

different fibration-types in F-theory compactifications.

Since the results of [17] apply to any fibration with two independent sections, it must

be possible to bring the factorised Tate models with just one U(1) symmetry into the

form (5.1). More precisely, the SU(5)×U(1)X and the SU(5)×U(1)PQ models of sections 3

and 4 should arise as further specialisations of (5.1) such as to account for the non-Abelian

gauge symmetry along w = 0.

In the case of the 4− 1 factorisation, it is indeed straightforward to provide the iden-

tification with (5.1). For C1, C2, C3 and B we only have to consider the definition of

the section that we gave in section 2.3.1 and match it with (5.2). To obtain also C0, we

use (5.1). The result of the identification is

B = c1,

C0 = 14 w

2 (d23 + 4wα),

C1 = 12 w (−c1 d3 d4 + 2w d2), (5.4)

C2 = 14 (c21 d

24 + 4w c0 d4 − 2w c1 d3),

C3 = w c0 + 12 c

21 d4 .

Along the same lines one can also match the coefficients of the 3−2 factorisation. The

only difference to the above case is that special care is required in identifying the section

because in 2.3.2 the section is given in terms of the (torus) sum of two points. Taking this

into account one finds

B = δ,

C0 = 14 w

2 (d23 α2 + 4wαγ),

C1 = 12 w (c2 d

23 α+ 2w (d2 α+ c2 γ)), (5.5)

C2 = 14 (c22 d

23 + 4w c2 d2 − 2w d3 α δ),

C3 = w β − 12 c2 d3 δ .

Indeed in both cases the base polynomials B,Ci are of a non-generic form. In particular

the powers of w are responsible for the SU(5) singularity along w = 0.

5.1 The SU(5) resolution

The hypersurface equation (5.3) with the coefficients (5.4) or (5.5) still exhibits an SU(5)-

singularity at w = 0. To resolve this singularity we find it more convenient to rewrite (5.3)

in a form where all monomials in the homogeneous coordinates of Bl[0,1,0]P[1,1,2] with bi-

degree (4, 3) show up. Equation (5.3) then becomes

B2 V2W + sW 2 +B1 sW V U+B0 s

2W U2 = C3 V3 U+

+C2s V2 U2 + C1 s

2 V U3 + C0 s3 U4 ,

(5.6)

– 27 –

JHEP03(2013)098

where the coefficients for the 4− 1 factorisation are given by

B0 = −w d3 = wB0,1,

B1 = c1 d4 = B1,0,

B2 = c1 = B2,0,

C0 = w3 α = w3C0,3, (5.7)

C1 = w2 d2 = w2C1,2,

C2 = w c0 d4 = wC2,1,

C3 = w c0 = wC3,1

and for the 3− 2 factorisation by

B0 = −w d3 α = wB0,1,

B1 = −c2 d3 = B1,0,

B2 = δ = B2,0,

C0 = w3 αγ = w3C0,3, (5.8)

C1 = w2 (d2α+ c2γ) = w2C1,2,

C2 = w c2 d2 = wC2,1,

C3 = w β = wC3,1.

To get back to (5.3), one just has to complete the square on the lefthand side of (5.6) and

do a coordinate redefinition.

The fibration (5.6) lends itself to a toric resolution of the singularity. From the classi-

fication of tops [67] one finds that for generic Bi,j and Ci,j there would be only an SU(4)

singularity at W = V = w = 0. We start with the resolution of this SU(4) singularity

which is in both cases, (5.7) and (5.8), the same. Using an approach similar to [12], which

is actually equivalent to the top constructions of [67], we find the ambient five-fold X5 of

table 2 and the proper transform of the hypersurface equation taking the form

e2e3B2,0 V2W + e1e2 sW

2 +B1,0 sW V U + e1 e0B0,1 s2W U2 = e2e

23 e0C3,1 V

3 U+

+ e3 e0C2,1s V2 U2 + e1e3 e

20C1,2 s

2 V U3 + e21e3 e30C0,3 s

3 U4.

(5.9)

This resolution allows for different Stanley-Reisner ideals. For brevity we use the following

Stanley-Reisner ideal in the sequel,

{V s, V e1, W U, W e0, W e3, U e2, s e2, e0 e2, U e3, s e0, s e3} , (5.10)

reserving a more systematic analysis also of the other triangulations for [68].

Due to the non-genericity of the coefficients, equation (5.9) is still singular. This follows

from the fact that it factorises, concretely for the 4− 1 model as

e1 sQ = V P1 P2 (5.11)

– 28 –

JHEP03(2013)098

V W U s e1 e2 e3 e0 PT

[w] · · · · · · · 1 ·K 1 2 · · · · · · 4

[B] · · · 1 · · · · 1

[U ] 1 2 1 · · · · · 4

[s] 1 1 · 1 · · · · 3

E1 · −1 · · 1 · · −1 −1

E2 −1 −2 · · · 1 · −1 −3

E3 −1 −1 · · · · 1 −1 −2

1 0 −1 −1 0 1 1 0

−1 1 −1 0 1 1 0 0

0 0 0 0 v v v v

Table 2. Divisor classes and coordinates of the ambient space with V , W , U , s the coordinates

of the “fibre ambient space” of the Calabi-Yau four-fold. Note that the base classes W , B and

K = c1(B) are included. The bottom of the table is only relevant if the entire 4-fold including the

base is torically embedded. It lists a choice for the vectors corresponding to the one-cones of the

toric fan.

with

Q = −e2W 2 + e0 d3 sW U2 + e3 e20 d2 s V U

3 + e1e3 e30 α s

2 U4,

P1 = e2e3 V + d4 sU, (5.12)

P2 = c1W − e3 e0 c0 V U

and for the 3− 2 model as

e2 Q = sU P1 P2 (5.13)

with

Q = e1 sW2 − e23 e0 β V 3 U + e3 δ V

2W,

P1 = e3 e0 d2 U V + d3W + e1e3 e20 γ sU

2, (5.14)

P2 = c2 V + e1 e0 α sU.

From (5.11) and (5.13) and the Stanely-Reisner ideal one easily observes that the excep-

tional divisors E1 = {e1 = 0} and E2 = {e2 = 0}, respectively, split into two on the

hypersurface. Since the two parts of the factorised divisor intersect each other, we obtain

the P1-structure of an SU(5)-singularity.

Due to the Stanley-Reisner ideal (5.10) only e1, Q, P1 and P2, for the 4−1 model, and

e2, Q, P1 and P2, for the 3− 2 model, have a common locus. Therefore, the last resolution

step is in both cases a small resolution given, respectively, by

λ1 e1 s = λ2 P2 , λ2Q = λ1 V P1 (5.15)

and

λ1 e2 = λ2 sP2 , λ2 Q = λ1 U P1 , (5.16)

– 29 –

JHEP03(2013)098

with λ1 and λ2 the homogeneous coordinates of some appropriate line bundle over the

blown-up ambient space X5. Therefore, we obtain in both cases an ambient six-fold, X6

and X6.

One may check that that the only possible remaining singularities in this procedure

are at the intersection of γ = α = 0. To be on the safe side we therefore restrict ourselves

to base spaces B where the intersection structure excludes this locus. A more detailed

investigation of such loci, if present, will be given in [68].

We can now examine the P1-structure in the resolved geometry. We start in co-

dimension one with w = 0. Here, as we mentioned already, the only difference to the

‘standard’ SU(5) case is that two of the P1’s in the fiber come from the splitting of one of

the P1’s of the SU(4) resolution. The fibration of these P1s over w = 0 give rise to a set of

divisors Ei intersecting like the Cartan of SU(5) if we adopt the respective labelings

E0 = [e0 ∩ PCI],

E1 = [e1 ∩ P1 ∩ λ2],E2 = [e1 ∩ P2 ∩ λ2Q− λ1 P1],

E3 = [e2 ∩ PCI],

E4 = [e3 ∩ PCI],

and

E0 = [e0 ∩ PCI],

E1 = [e1 ∩ PCI],

E2 = [e2 ∩ P1 ∩ λ2],E3 = [e2 ∩ P2 ∩ λ2 Q− λ1 P1],

E4 = [e3 ∩ PCI],

(5.17)

where PCI and PCI refer to the complete intersection (5.15) and (5.16), respectively.

5.2 The singlet curves in the resolved P[1,1,2]-fibration

It is straightforward to re-analyse the structure of the fibers over the matter curves and

the Yukawa interaction points starting from the resolved 4-folds (5.15) and (5.16). While

we relegate a presentation of the details of this analysis to [68] let us merely note that the

findings of sections 3 and 4 are indeed confirmed except for slight details in the structure

of the Yukawa points.11 Concerning the analysis of the co-dimension three singularties we

point out that at the 5(1)−25

(2)−315 point we find a fiber of extended A6 type.

What we do present now is an analysis of the charged singlet curves, which as discussed

in section 4 is subtle in the small resolution approach of the previous sections. Indeed, the

structure of U(1) charged singlets in the resolved P[1,1,2] model (5.6) has been worked out in

detail in [17] for a fibration over the base P2. Since the appearance and further resolution

of the SU(5) singularity over w = 0 is irrelevant for the generic points on the singlet curves,

we can adopt this analysis. According to the general logic of [17], it then follows that the

singlets of U(1) charge ±10 localise in the fibre over the curve

B = C3 = 0 . (5.18)

The charge ±5 singlets are located at the generic intersection of the two loci

0 = −12 B

4 C1 +B2 C2 C3 − C33,

0 = −B6 C0 +B4 C22 − 2B2 C2 C2

3 + C43

(5.19)

11These, however, change for different triangulations anyway.

– 30 –

JHEP03(2013)098

charge 10 charge 5

Figure 3. Schematic drawing of the fibre over the charge 10 and 5 singlet curves. The green and

blue crosses indicate the intersections with Z and S, respectively. In the case of the charge 10

singlet, S becomes one of the P1’s, which we indicate by the blue ball.

where B and C3 do not simultaneously vanish.

To read off the charges, we recall that these were given by the intersection of the P1’s

of the resolved singularity with the divisor

wX = 5(S − Z . . .),

where the ellipsis indicates terms irrelevant for the current explanation. In [17] it was now

shown that at the locus (5.18) equation (5.3) becomes

sD = 0 ,

whereas at (5.19) it factorises as

(A−B)(A+B) = 0 ,

where A, B and D are some polynomials. Therefore, the two P1’s into which the torus

factorises are in the one case

P1s : s = 0 and P1

D : D = 0

and in the other case

P1A− : A−B = 0 and P1

A+ : A+B = 0.

To calculate now the charge for P1D we observe from figure 3 that it intersects S two times

but does not have any overlap with Z, since Z intersects only P1s. Hence, we obtain +10

for∫P1D

wX . As was already explained in the discussion of the singlets in sections 3 and 4,

the M2-brane wrapping the second P1 is just the adjoint state to the M2-brane wrapping

the other one. Therefore, P1s must have charge −10. From figure 3 it is also clear that∫

P1A−

wX = −∫P1A+

wX = 5. Again M2-branes wrapping P1A− and P1

A+ are adjoint states

to each other.

– 31 –

JHEP03(2013)098

In the 4−1 case, a simultaneous vanishing of c1 and c0 is forbidden because this would

lead to non-Kodaira singularities as stated after eq. (2.28). Therefore, there are no singlets

1±10 but only 1±5 states localized on the curve

0 = −12w

2(c21 (c21 d2 + c0 c1 d3 + c20 d4) + 2 c30w),

0 = w3 (c20 (c31 d3 + c20w) + c61 α),(5.20)

in agreement with the results of section 3. For the 3 − 2 model the situation is different

because now we also have charge ±10 singlets at

δ = β = 0 , (5.21)

besides the charge five singlets at

0 = −12 w(

2w2 β3 + c22 d3 δ2 (d3 β + d2 δ) + w δ

(α δ2 (d3 β + d2 δ)+

+ c2 (γ δ3 − 3 d3 β2 − 2 d2 β δ)

))0 = w2

(w2 β4 + c2 δ

2 (d3 β + d2 δ) (c2 d3 β + c2 d2 δ − d3 α δ2)+

− w δ(αγ δ5 − d3 αβ2 δ2 + 2 c2 β

2 (d3 β + d2 δ)))

(5.22)

and δ 6= 0 or β 6= 0.

Note again that we are explicitly excluding the locus α = γ = 0, where potential

singularities may remain. From the arguments presented in section 4 it is expected that

this locus gives rise to charge ±10 singlets as well, but a verification of this conjecture via

an explicit resolution of this locus is reserved for [68].

6 Summary

In this article we have studied 4-dimensional F-theory compactifications with U(1) sym-

metries in addition to a non-Abelian gauge group G, taken to be SU(5) for definiteness.

We developed a systematic approach to construct such models as factorised Tate models

with multiple sections. We have provided the form of the factorised Tate models for a wide

range of possible U(1) symmetries and exemplified the resolution of the associated singular

fibrations for the two cases with a single U(1) — called SU(5)×U(1)X and SU(5)×U(1)PQ.

These can be treated either patchwise by a small resolution procedure or in terms of the

resolution of a P[1,1,2] fibration as in [17]. Our results obtained in both approaches con-

cerning the fiber structure agree. An explicit construction of the U(1) generators after

the resolution and analysis of the intersection with the fibers over the matter curves has

allowed us to derive the Abelian charges of all matter states directly from the geometry.

As in [10–13] this also provides us with the associated U(1) flux for chiral model building.

The SU(5)×U(1)X model is a generalization of the U(1) restricted Tate model [10–13]

and all the factorised Tate models flow, in the vicinity of the SU(5) divisor, to the split

spectral models [19–22, 24]. What makes an analysis of Abelian gauge groups within a fully

global treatment of the geometry so crucial is the fact that U(1) symmetries are sensitive

– 32 –

JHEP03(2013)098

to geometric details away from the SU(5) divisor. In particular, charged singlets arise from

curves extending into the bulk of the compactification space, where local methods fail. As

one of the surprises we have encountered, in the global version of the SU(5)×U(1)PQ model

an extra set of charged singlets of charge ±10 appears, which apparently do not follow from

a decomposition of the 248 of a single E8 gauge group. Since these states couple to the

SU(5) matter these novel states can in principle influence the phenomenology of the model.

The extension of our methods to a detailed analysis of the models with several Abelian

factors as classified in this paper is under way [68].

We hope that the systematic construction of global F-theory GUT models exhibiting

additional U(1) symmetries, without restricting the matter spectrum or Yukawa couplings,

outlined in this paper will open the way to realising much of the successful phenomenology

of local F-theory models in a fully global setting.

Acknowledgments

We thank Max Kerstan for initial collaboration and Harald Skarke for very useful discus-

sions. We also thank Jan Borchmann for providing the details of the A3 top and Kang-Sin

Choi for discussion. CM thanks IPhT CEA Saclay, especially Mariana Grana, for hospital-

ity. Furthermore, we are grateful to David Morrison and Daniel Park for providing us with

a draft of their paper. The research of EP is supported by a Marie Curie Intra European

Fellowship within the 7th European Community Framework Programme. The work of CM

and TW was funded in part by the DFG under Transregio TR 33 “The Dark Universe”.

A More factorised Tate models

In this appendix we give the form of (2.14) explicitly for the other possible facorisations.

We use here the notation of [24] and in particular denote products ci . . . cj = ci...j .

A.1 2 − 2 − 1 factorisation

The factorisation of PT |X=0 before the SU(5) resolution is(c1t

2 + c2t+ c3) (c4t

2 + c5t+ c6)

(c7t+ c8) . (A.1)

The ai,n are given by

a6,5 = c368 ,

a5,4 = c367 + c358 + c268 ,

a4,3 = c357 + c267 + c348 + c258 + c168 ,

a3,2 = c347 + c257 + c167 + c248 + c158 ,

a2,1 = c247 + c157 + c148 ,

a1 = c147 . (A.2)

– 33 –

JHEP03(2013)098

A solution to the tracelessness constraint was given in [22] and reads

c3 = αβδ1 ,

c2 = γδ1 ,

c6 = αβεδ2 ,

c5 = −δ2 (γε+ c7β) ,

c8 = αε . (A.3)

Unlike the case of a single U(1) there now appears a subtlety in defining this solution

because a5,4 is a sum of 3 terms but can be set to vanish with only 2 sections. So for

example setting ε = β = 0 solves a5,4 = 0 without imposing c3 = 0 necessarily, even

though the solution (A.3) would constrain it to be so. There are a number of such special

cases that occur on the intersection locus of 2 sections. Therefore placing constraints on

the intersection numbers in order to avoid non-Kodaira singularities is very complicated,

since the solution (A.3) could be adjusted accordingly on these special loci to avoid such

a bad singularity. Hence the solution (A.3) is not the most general one. In this paper we

will not perform an analysis of the most general solution and the resulting constraints on

intersection numbers to avoid non-Kodaira singularities. We will work explicitly with the

solution (A.3) and leave the most general analysis for future work.

With this solution, in order to ensure no non-Kodaira singularities, for generic sections

we should impose that the following intersections must vanish

α · c7 , ε · c7 , c1 · δ , c4 · δ . (A.4)

Since the sections are now objects over the whole base we should also consider triple

intersections which must vanish

c1 · α · γ , c1 · β · γ , c4 · γ · ε , c4 · c7 · β . (A.5)

Within the patch e1 = e2 = e4 = 1 we have that the Tate polynomial, after the SU(5)

resolution, can be written as (2.14) with

Y1 = c7t+ αe0εz , (A.6)

Y2 = c1t2 + δ1e0γtz + αβδ1e

20z

2 ,

Y3 = c4t2 − βc7δ2e0tz − δ2e0εγtz + αβδ2e

20εz

2 ,

X = t2 − x ,Q = e3x

2 + c1c4c7t3z + c1c4c7txz − βc1c27δ2e0t2z2 + αc1c4e0εt

2z2 + c4c7δ1e0γt2z2

−c1c7δ2e0εγt2z2 − βc1c27δ2e0xz2 + αc1c4e0εxz2 + c4c7δ1e0γxz

2 − c1c7δ2e0εγxz2

+αβc4c7δ1e20tz

3 − βc27δ1δ2e20γtz3 + αc4δ1e20εγtz

3 − αc1δ2e20ε2γtz3 − c7δ1δ2e20εγ2tz3

−αβ2c27δ1δ2e30z4+α2βc4δ1e30εz

4+α2βc1δ2e30ε

2z4−αβc7δ1δ2e30εγz4−αδ1δ2e30ε2γ2z4 .

In the case of multiple U(1)s the single small resolution (2.30) of course is not sufficient

to completely resolve the manifold but a generalisation of it is required. The resolution

of this particular type of binomial singularity was studied in detail in [43]. We introduce

– 34 –

JHEP03(2013)098

two new P1s spanned by {λ1, λ2} and {σ1, σ2}, in terms of which the resolved four-fold is

given by

Y4 : {Y1λ2 = Qλ1} ∩ {Y2σ2 = Xσ1} ∩ {λ2σ2 = λ1σ1Y3} . (A.7)

There are 6 different possible resolutions, related by flop transitions, corresponding to

permuting {Y1, Y2, Y3} in (A.7).

A.2 3 − 1 − 1 factorisation

In this case we have the factorisation of PT |X=0 as(c1t

3 + c2t2 + c3t+ c4

)(c5t+ c6) (c7t+ c8) . (A.8)

The ai,n are given by

a6,5 = c468 ,

a5,4 = c467 + c458 + c368 ,

a4,3 = c457 + c367 + c358 + c268 ,

a3,2 = c357 + c267 + c258 + c168 ,

a2,1 = c257 + c167 + c158 ,

a1 = c157 . (A.9)

A possible solution to the tracelessness constraint was given in [22] and reads

c6 = αβ ,

c8 = αγ ,

c4 = αβγδ ,

c3 = −δ (c5γ + βc7) . (A.10)

With this solution, in order to ensure no non-Kodaira singularities, for generic sections we

should impose that the following intersections must vanish

c5 · α , c7 · α , c5 · β , c7 · γ , c1 · c2 · δ . (A.11)

Note that again, as discussed in section A.1, the solution (A.10) is not the most general

one.

Within the patch e1 = e2 = e4 = 1 we have that the Tate polynomial can be written

as (2.14) with

Y1 = c5t+ αβe0z ,

Y2 = c7t+ αe0γz ,

Y3 = c1t3 + c2e0t

2z − βc7δe20tz2 − c5δe20γtz2 + αβδe30γz3 ,

X = t2 − x ,Q = e3x

2 + c1c5c7t3z + c1c5c7txz + αβc1c7e0t

2z2 + c2c5c7e0t2z2 + αc1c5e0γt

2z2

+αβc1c7e0xz2 + c2c5c7e0xz

2 + αc1c5e0γxz2 + αβc2c7e

20tz

3 − βc5c27δe20tz3

+α2βc1e20γtz

3 + αc2c5e20γtz

3 − c25c7δe20γtz3 − αβ2c27δe30z4

+α2βc2e30γz

4 − αβc5c7δe30γz4 − αc25δe30γ2z4 . (A.12)

– 35 –

JHEP03(2013)098

A.3 2 − 1 − 1 − 1 factorisation

The factorisation is (c1t

2 + c2t+ c3)

(c4t+ c7) (c5t+ c8) (c6t+ c9) . (A.13)

The ai,n are given by

a6,5 = c3789 ,

a5,4 = c2789 + c3678 + c3579 + c3489 ,

a4,3 = c1789 + c2678 + c2579 + c2489 + c3567 + c3468 + c3459 ,

a3,2 = c3456 + c1678 + c1579 + c1489 + c2567 + c2468 + c2459 ,

a2,1 = c2456 + c1567 + c1468 + c1459 ,

a1 = c1456 . (A.14)

A solution to the tracelessness constraint is

c7 = δα1 ,

c8 = δα2 ,

c9 = δα3 ,

c3 = δεα1α2α3 ,

c2 = −ε (c6α1α2 + c5α1α3 + c4α2α3) . (A.15)

The intersections which must vanish are

c4 · δ , c5 · δ , c6 · δ , c4 · α1 , c5 · α2 , c6 · α3 , c1 · ε ,c1 · α1 · α2 , c1 · α1 · α3 , c1 · α2 · α3 ,

c1 · c6 · α3 , c1 · c5 · α2 , c1 · c4 · α1 . (A.16)

Within the patch e1 = e2 = e4 = 1 we have that the Tate polynomial can be written

as (2.14) with

Y1 = c4t+ α1δe0z ,

Y2 = c5t+ α2δe0z ,

Y3 = c6t+ α3δe0z ,

Y4 = c1t2 − α2α3c4e0εtz − α1α3c5e0εtz − α1α2c6e0εtz + α1α2α3δe

20εz

2 ,

X = t2 − x ,Q = e3x

2 + c1c4c5c6t3z + c1c4c5c6txz + α3c1c4c5δe0t

2z2 + α2c1c4c6δe0t2z2

+α1c1c5c6δe0t2z2 − α2α3c

24c5c6e0εt

2z2 − α1α3c4c25c6e0εt

2z2 − α1α2c4c5c26e0εt

2z2

+α3c1c4c5δe0xz2 + α2c1c4c6δe0xz

2 + α1c1c5c6δe0xz2 − α2α3c

24c5c6e0εxz

2

−α1α3c4c25c6e0εxz

2 − α1α2c4c5c26e0εxz

2 + α2α3c1c4δ2e20tz

3

+α1α3c1c5δ2e20tz

3 + α1α2c1c6δ2e20tz

3 − α2α23c

24c5δe

20εtz

3 − α1α23c4c

25δe

20εtz

3

−α22α3c

24c6δe

20εtz

3 − 2α1α2α3c4c5c6δe20εtz

3 − α21α3c

25c6δe

20εtz

3

−α1α22c4c

26δe

20εtz

3 − α21α2c5c

26δe

20εtz

3 + α1α2α3c1δ3e30z

4

−α22α

23c

24δ

2e30εz4 − α1α2α

23c4c5δ

2e30εz4 − α2

1α23c

25δ

2e30εz4 −

α1α22α3c4c6δ

2e30εz4 − α2

1α2α3c5c6δ2e30εz

4 − α21α

22c

26δ

2e30εz4 . (A.17)

– 36 –

JHEP03(2013)098

A.4 1 − 1 − 1 − 1 − 1 factorisation

The factorisation is

(c1t+ c6) (c2t+ c7) (c3t+ c8) (c4t+ c9) (c5t+ c10) . (A.18)

The ai,n are given by

a6,5 = c10c6c7c8c9 ,

a5,4 = c10c4c6c7c8 + c10c3c6c7c9 + c10c2c6c8c9 + c1c10c7c8c9 + c5c6c7c8c9 ,

a4,3 = c10c3c4c6c7 + c10c2c4c6c8 + c1c10c4c7c8 + c4c5c6c7c8 + c10c2c3c6c9

+c1c10c3c7c9 + c3c5c6c7c9 + c1c10c2c8c9 + c2c5c6c8c9 + c1c5c7c8c9 ,

a3,2 = c10c2c3c4c6 + c1c10c3c4c7 + c3c4c5c6c7 + c1c10c2c4c8 + c2c4c5c6c8

+c1c4c5c7c8 + c1c10c2c3c9 + c2c3c5c6c9 + c1c3c5c7c9 + c1c2c5c8c9 ,

a2,1 = c1c10c2c3c4 + c2c3c4c5c6 + c1c3c4c5c7 + c1c2c4c5c8 + c1c2c3c5c9 ,

a1 = c1c2c3c4c5 . (A.19)

A solution to the tracelessness constraint is

c7 = δα1 ,

c8 = δα2 ,

c9 = δα3 ,

c10 = δα4 ,

c6 = δα1α2α3α4 ,

c1 = − (c5α1α2α3 + c4α1α2α4 + c3α1α3α4 + c2α2α3α4) . (A.20)

There are many intersections which must vanish for this solution to hold generally, most

notably αi · αj . Here, because of the strong constraints on intersection numbers, the

discussion in section A.1 regarding the fact that the solution is not the most general one

possible becomes even more crucial. We proceed with analysing the solution presented since

for the purposes of this paper it serves as a useful illustration of the general procedure,

but keep in mind that studying more general solutions to a5,4 = 0 in this case is of great

importance.

Within the patch e1 = e2 = e4 = 1 we have that the Tate polynomial can be written

as (2.14) with

Y1 = c2t+ α1δe0z ,

Y2 = c3t+ α2δe0z ,

Y3 = c4t+ α3δe0z ,

Y4 = c5t+ α4δe0z ,

Y5 = −α2α3α4c2t− α1α3α4c3t− α1α2α4c4t− α1α2α3c5t+ α1α2α3α4δe0z ,

X = t2 − x ,

– 37 –

JHEP03(2013)098

Q = e3x2 − α2α3α4c

22c3c4c5t

3z − α1α3α4c2c23c4c5t

3z − α1α2α4c2c3c24c5t

3z

−α1α2α3c2c3c4c25t

3z − α2α3α4c22c3c4c5txz − α1α3α4c2c

23c4c5txz

−α1α2α4c2c3c24c5txz − α1α2α3c2c3c4c

25txz − α2α3α

24c

22c3c4δe0t

2z2

−α1α3α24c2c

23c4δe0t

2z2 − α1α2α24c2c3c

24δe0t

2z2 − α2α23α4c

22c3c5δe0t

2z2

−α1α23α4c2c

23c5δe0t

2z2 − α22α3α4c

22c4c5δe0t

2z2 − 3α1α2α3α4c2c3c4c5δe0t2z2

−α21α3α4c

23c4c5δe0t

2z2 − α1α22α4c2c

24c5δe0t

2z2 − α21α2α4c3c

24c5δe0t

2z2

−α1α2α23c2c3c

25δe0t

2z2 − α1α22α3c2c4c

25δe0t

2z2 − α21α2α3c3c4c

25δe0t

2z2

−α2α3α24c

22c3c4δe0xz

2 − α1α3α24c2c

23c4δe0xz

2 − α1α2α24c2c3c

24δe0xz

2

−α2α23α4c

22c3c5δe0xz

2 − α1α23α4c2c

23c5δe0xz

2 − α22α3α4c

22c4c5δe0xz

2

−3α1α2α3α4c2c3c4c5δe0xz2 − α2

1α3α4c23c4c5δe0xz

2 − α1α22α4c2c

24c5δe0xz

2

−α21α2α4c3c

24c5δe0xz

2 − α1α2α23c2c3c

25δe0xz

2 − α1α22α3c2c4c

25δe0xz

2

−α21α2α3c3c4c

25δe0xz

2 − α2α23α

24c

22c3δ

2e20tz3 − α1α

23α

24c2c

23δ

2e20tz3

−α22α3α

24c

22c4δ

2e20tz3 − 2α1α2α3α

24c2c3c4δ

2e20tz3 − α2

1α3α24c

23c4δ

2e20tz3

−α1α22α

24c2c

24δ

2e20tz3 − α2

1α2α24c3c

24δ

2e20tz3 − α2

2α23α4c

22c5δ

2e20tz3

−2α1α2α23α4c2c3c5δ

2e20tz3 − α2

1α23α4c

23c5δ

2e20tz3 − 2α1α

22α3α4c2c4c5δ

2e20tz3

−2α21α2α3α4c3c4c5δ

2e20tz3 − α2

1α22α4c

24c5δ

2e20tz3 − α1α

22α

23c2c

25δ

2e20tz3

−α21α2α

23c3c

25δ

2e20tz3 − α2

1α22α3c4c

25δ

2e20tz3 − α2

2α23α

24c

22δ

3e30z4

−α1α2α23α

24c2c3δ

3e30z4 − α2

1α23α

24c

23δ

3e30z4 − α1α

22α3α

24c2c4δ

3e30z4

−α21α2α3α

24c3c4δ

3e30z4 − α2

1α22α

24c

24δ

3e30z4 − α1α

22α

23α4c2c5δ

3e30z4

−α21α2α

23α4c3c5δ

3e30z4 − α2

1α22α3α4c4c5δ

3e30z4 − α2

1α22α

23c

25δ

3e30z4 . (A.21)

B Relation to other approaches to U(1)s

B.1 Relation to the U(1)-restricted Tate model

In [10] a method for constructing elliptic fibrations that support a global U(1) symmetry

was proposed. In this appendix we discuss the relation of this method to the results

discussed in this paper. The model of [10] corresponds to the 4− 1 factorisation but with

the added constraint that c1 = 1 [19] so that the 10-matter curve C10(2) in (3.3) is switched

off. It was shown that after an appropriate coordinate transformation

x→ x+ (wc0z)2 , y → y − (wc0z)

3 , (B.1)

the monomial associated to a6 in (2.2) vanishes. In that case it was argued using Tate’s

algorithm that after the transformation there is an SU(2) singularity over the curve

a4,3 = a3,2 = 0 . (B.2)

This singularity can then be resolved by a blow-up x→ xs, y → ys which accounts for the

additional U(1) (see also [12, 13]). The SU(2) singularity was also identified in a different

– 38 –

JHEP03(2013)098

way in [11] by moving to the Sen coordinates in which case the Weiestrass polynomial takes

the form

Y−Y+ −XQ = z6a6 , (B.3)

and so in a coordinate basis where a6 = 0 takes the form of a conifold. The singularity

locus in the Sen coordinates Y+ = Y− = X = Q = 0 coincided with locus determined from

Tate’s algorithm x = y = a4,3 = a3,2 = 0.

It is possible to generalise this approach to understanding the U(1)s to other cases

as follows. We consider the special case where the U(1) is associated to a section which

can be written in the form x = A2 and y = −A3, with A and B being some holomorphic

polynomials. Since we have been discussing sections that satisfy y2 = x3 the constraint

is that the additional holomorphic equation specifying the section can be written in the

form x = A2. If this is possible, then the procedure employed in the U(1)-restricted model

can be applied generally. The idea is to shift the coordinates by the section x → x + A2,

y → y − A3 and in the new coordinates it must be that a6 = 0 since at x = y = 0 we

recover the section that satisfies PT = a6z6 = 0. Once this coordinate choice is made the

singularity can be identified using the two methods described above.

Therefore the particular case studied in [10] was applying this procedure to a fibration

with a section satisfying the above constraints with A = c0wz. This section is related to

the general section for the 4 − 1 case which is identified in (2.29) as one of the factors Yito be

Ac1 = c0wz , (B.4)

where we parameterically solved the y2 = x3 part of the section by setting t = y/x = −A.

Now we see that A is only holomorphic if we set c1 = 1 and so turning off one of the

10-matter curves was crucial to the success of the procedure. Generally however A is

only meromorphic and diverges on the second 10-matter curve and where this procedure

breaks down. For this more general case the approach described in this paper must be

adopted. It is possible to check that if we continue with the U(1)-restricted procedure

without worrying about the meromorphicity in c1 the singularity locus identified using

Tate’s algorithm (B.2) or using the Sen coordinates (B.3) both match the singularity locus

obtained using our procedure (2.29).

B.2 Relation to split spectral cover models

In the local limit the split Tate model flows to the split spectral cover construction. The

local limit is well defined before the resolution of the SU(5) singularity and corresponds to

taking w → 0. The local limit of the Tate model (2.11) was studied in [44, 56, 57] where

the section (2.5) was termed the Tate divisor. In order to recover the spectral cover the

limit must be taken such that also t→ 0 while keeping the ratio finite [56, 57]

w → 0 ,w

t→ s , (B.5)

in the patch z = 1. After the proper transform of dividing out by the overall factor of t5

we recover the Higgs bundle on SGUT

b5 + b4s+ b3s2 + b2s

3 + b0s5 = 0 . (B.6)

– 39 –

JHEP03(2013)098

Here we denote

a1|w=0 = b5 , a2,1|w=0 = b4 , a3,2|w=0 = b3 , a4,3|w=0 = b2 , a6,5|w=0 = b0 . (B.7)

We should think of this in terms of an underlying E8 symmetry broken according to E8 →SU(5)GUT×SU(5)⊥ in two equivalent ways. Either through an 8-dimensional gauge theory

on the GUT brane with gauge group E8 that is broken to SU(5)GUT by a spatially varying

adjoint Higgs field ϕ with vev in the SU(5)⊥, the precise map being [18]

b1 = Tr [ϕ] , b2 = −1

2Tr[ϕ2], b5 = det [ϕ] . (B.8)

The other way is through an A4 singularity, corresponding to SU(5)⊥, that is fibered over

SGUT [1–4]. A fully deformed A4 singularity takes the form

y2 = x2 +5∏

i=1

(s+ ti) , (B.9)

where the ti are 5 deformation parameters, which are functions on SGUT, that can be

explicitly mapped to the Cartan U(1)s inside SU(5)⊥. The proper identification with the

bi is simply the expansion

5∏i=1

(s+ ti) =

(b5b0

)+

(b4b0

)s+

(b3b0

)s2 +

(b2b0

)s3 + s5 . (B.10)

This determines the bi as the elementry symmetric polynomials in the ti.

The A4 singularity has a Weyl group action which interchanges the ti so as to preserve

the bi. Generally the fibration over SGUT can act with this group which in F-theory

is termed monodromies [18, 60]. More generally we can think of the Higgs bundle as

taking value in various subgroups of SU(5)⊥ that preserve some U(1) symmetries, and in

diagonalising the Higgs so that the map (B.8) holds, branch cuts are induced in the form

of the ti as functions on SGUT which map them to each other as we move around the

branch [61]. The case where the Higgs preserves the full Cartan of SU(5)⊥ ⊃ S[U(1)5

]is

mapped to the case where there are no monodromies, while maintaining smaller Abelian

subgroups corresponds to non-trivial monodromies.

This maps directly to the product structure of (B.10), where we see that under no

identification of the ti, (B.6) factorises into 5 factors. Each factor corresponds to a U(1)

with a tracelessness constraint b1 = 0 leaving the 4 Cartan U(1)s as linearly indepenedent.

As we identify the ti (B.6) decomposes into fewer factors implying fewer U(1)s and finally

if the fibration uses the full Weyl group there is no splitting at all and no U(1)s.

Exactly this structure is what is termed a split spectral cover, where we simply com-

pactify the surface (B.10) by writing s in terms of homogenous coordinates s = U/V [18].

The discussion presented is the local understanding of the required splitting structure

of (B.6) in order to preserve a U(1). So for example the 4− 1 factorisation is such that

(c0s+ c1)(s4d0 + s3d1 + s2d2 + sd3 + d4

)= b5 + b4s+ b3s

2 + b2s3 + b0s

5 , (B.11)

– 40 –

JHEP03(2013)098

where the ci and di are holomorphic functions on SGUT. This fixes the form of the bi and

imposes a tracelessness constraint on the ci and di.

Note that, of course, the local splitting is a weaker constraint on the full bi than a

factorised Tate model which also constrains the w dependence of the bi. For example,

it was shown in [55] that for the case of a Heterotic dual there are constraints on the

complex structure moduli of the F-theory CY which manifest in the specific form of the w

dependence of the bi. These higher order terms in the bi precisely take the form so as to

respect the appropriate factorisation structure (2.29) which means they can be written in

terms of higer order terms in the ci and di, specifically

d0 = d0|w=0 − Fwc0 , d1 = d1|w=0 + Fwc1 , (B.12)

where F is some arbitrary function.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License which permits any use, distribution and reproduction in any medium,

provided the original author(s) and source are credited.

References

[1] C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory - I, JHEP

01 (2009) 058 [arXiv:0802.3391] [INSPIRE].

[2] C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory - II:

experimental predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].

[3] R. Donagi and M. Wijnholt, Model building with F-theory, Adv. Theor. Math. Phys. 15

(2011) 1237 [arXiv:0802.2969] [INSPIRE].

[4] R. Donagi and M. Wijnholt, Breaking GUT groups in F-theory, Adv. Theor. Math. Phys. 15

(2011) 1523 [arXiv:0808.2223] [INSPIRE].

[5] C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].

[6] F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [INSPIRE].

[7] T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav.

27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].

[8] D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl.

Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].

[9] D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2., Nucl.

Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].

[10] T.W. Grimm and T. Weigand, On abelian gauge symmetries and proton decay in global

F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [INSPIRE].

[11] A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl.

Phys. B 856 (2012) 129 [arXiv:1107.5337] [INSPIRE].

[12] S. Krause, C. Mayrhofer and T. Weigand, G4 flux, chiral matter and singularity resolution in

F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].

– 41 –

JHEP03(2013)098

[13] T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP

03 (2012) 027 [arXiv:1111.1232] [INSPIRE].

[14] S. Krause, C. Mayrhofer and T. Weigand, Gauge fluxes in F-theory and type IIB orientifolds,

JHEP 08 (2012) 119 [arXiv:1202.3138] [INSPIRE].

[15] J. Marsano, H. Clemens, T. Pantev, S. Raby and H.-H. Tseng, A global SU(5) F-theory

model with Wilson line breaking, JHEP 01 (2013) 150 [arXiv:1206.6132] [INSPIRE].

[16] M. Cvetic, T.W. Grimm and D. Klevers, Anomaly cancellation and abelian gauge symmetries

in F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].

[17] D.R. Morrison and D.S. Park, F-theory and the Mordell-Weil group of elliptically-fibered

Calabi-Yau threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].

[18] R. Donagi and M. Wijnholt, Higgs bundles and UV completion in F-theory,

arXiv:0904.1218 [INSPIRE].

[19] J. Marsano, N. Saulina and S. Schafer-Nameki, Monodromies, fluxes and compact

three-generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [INSPIRE].

[20] J. Marsano, N. Saulina and S. Schafer-Nameki, Compact F-theory GUTs with U(1) (PQ),

JHEP 04 (2010) 095 [arXiv:0912.0272] [INSPIRE].

[21] M.J. Dolan, J. Marsano and S. Schafer-Nameki, Unification and phenomenology of F-theory

GUTs with U(1)PQ, JHEP 12 (2011) 032 [arXiv:1109.4958] [INSPIRE].

[22] M.J. Dolan, J. Marsano, N. Saulina and S. Schafer-Nameki, F-theory GUTs with U(1)

symmetries: generalities and survey, Phys. Rev. D 84 (2011) 066008 [arXiv:1102.0290]

[INSPIRE].

[23] E. Palti, A note on hypercharge flux, anomalies and U(1)s in F-theory GUTs,

arXiv:1209.4421 [INSPIRE].

[24] E. Dudas and E. Palti, On hypercharge flux and exotics in F-theory GUTs, JHEP 09 (2010)

013 [arXiv:1007.1297] [INSPIRE].

[25] E. Dudas and E. Palti, Froggatt-Nielsen models from E8 in F-theory GUTs, JHEP 01 (2010)

127 [arXiv:0912.0853] [INSPIRE].

[26] S. King, G. Leontaris and G. Ross, Family symmetries in F-theory GUTs, Nucl. Phys. B

838 (2010) 119 [arXiv:1005.1025] [INSPIRE].

[27] C.-M. Chen and Y.-C. Chung, Flipped SU(5) GUTs from E8 singularities in F-theory, JHEP

03 (2011) 049 [arXiv:1005.5728] [INSPIRE].

[28] E. Kuflik and J. Marsano, Comments on flipped SU(5) (and F-theory), JHEP 03 (2011) 020

[arXiv:1009.2510] [INSPIRE].

[29] G. Leontaris and G. Ross, Yukawa couplings and fermion mass structure in F-theory GUTs,

JHEP 02 (2011) 108 [arXiv:1009.6000] [INSPIRE].

[30] C.-M. Chen and Y.-C. Chung, On F-theory E6 GUTs, JHEP 03 (2011) 129

[arXiv:1010.5536] [INSPIRE].

[31] J. Marsano, Hypercharge flux, exotics and anomaly cancellation in F-theory GUTs, Phys.

Rev. Lett. 106 (2011) 081601 [arXiv:1011.2212] [INSPIRE].

[32] C. Ludeling, H.P. Nilles and C.C. Stephan, The potential fate of local model building, Phys.

Rev. D 83 (2011) 086008 [arXiv:1101.3346] [INSPIRE].

– 42 –

JHEP03(2013)098

[33] K.-S. Choi and B. Kyae, Weak mixing angle and proton stability in F-theory GUT, Nucl.

Phys. B 855 (2012) 1 [arXiv:1102.0591] [INSPIRE].

[34] J.C. Callaghan, S.F. King, G.K. Leontaris and G.G. Ross, Towards a realistic F-theory GUT,

JHEP 04 (2012) 094 [arXiv:1109.1399] [INSPIRE].

[35] E. Palti, Wavefunctions and the point of E8 in F-theory, JHEP 07 (2012) 065

[arXiv:1203.4490] [INSPIRE].

[36] I. Antoniadis and G. Leontaris, Building SO(10) models from F-theory, JHEP 08 (2012) 001

[arXiv:1205.6930] [INSPIRE].

[37] J.C. Callaghan and S.F. King, E6 models from F-theory, arXiv:1210.6913 [INSPIRE].

[38] H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, More on dimension-4 proton decay

problem in F-theory — Spectral surface, discriminant locus and monodromy, Nucl. Phys. B

840 (2010) 304 [arXiv:1004.3870] [INSPIRE].

[39] M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov et al., Geometric

singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215

[hep-th/9605200] [INSPIRE].

[40] S. Katz, D.R. Morrison, S. Schafer-Nameki and J. Sully, Tate’s algorithm and F-theory,

JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].

[41] R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl.

Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].

[42] T.W. Grimm, S. Krause and T. Weigand, F-theory GUT vacua on compact Calabi-Yau

fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [INSPIRE].

[43] M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, arXiv:1107.0733

[INSPIRE].

[44] J. Marsano and S. Schafer-Nameki, Yukawas, G-flux and spectral covers from resolved

Calabi-Yau’s, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].

[45] C.-M. Chen, J. Knapp, M. Kreuzer and C. Mayrhofer, Global SO(10) F-theory GUTs, JHEP

10 (2010) 057 [arXiv:1005.5735] [INSPIRE].

[46] A. Collinucci and R. Savelli, On flux quantization in F-theory, JHEP 02 (2012) 015

[arXiv:1011.6388] [INSPIRE].

[47] J. Knapp, M. Kreuzer, C. Mayrhofer and N.-O. Walliser, Toric construction of global

F-theory GUTs, JHEP 03 (2011) 138 [arXiv:1101.4908] [INSPIRE].

[48] A. Collinucci and R. Savelli, On flux quantization in F-theory II: unitary and symplectic

gauge groups, JHEP 08 (2012) 094 [arXiv:1203.4542] [INSPIRE].

[49] M. Kuntzler and S. Schafer-Nameki, G-flux and spectral divisors, JHEP 11 (2012) 025

[arXiv:1205.5688] [INSPIRE].

[50] R. Tatar and W. Walters, GUT theories from Calabi-Yau 4-folds with SO(10) singularities,

arXiv:1206.5090 [INSPIRE].

[51] T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive abelian gauge symmetries and

fluxes in F-theory, JHEP 12 (2011) 004 [arXiv:1107.3842] [INSPIRE].

[52] G. Aldazabal, A. Font, L.E. Ibanez and A. Uranga, New branches of string compactifications

and their F-theory duals, Nucl. Phys. B 492 (1997) 119 [hep-th/9607121] [INSPIRE].

– 43 –

JHEP03(2013)098

[53] P. Candelas, E. Perevalov and G. Rajesh, Comments on A, B, c chains of heterotic and

type-II vacua, Nucl. Phys. B 502 (1997) 594 [hep-th/9703148] [INSPIRE].

[54] P. Berglund, A. Klemm, P. Mayr and S. Theisen, On type IIB vacua with varying coupling

constant, Nucl. Phys. B 558 (1999) 178 [hep-th/9805189] [INSPIRE].

[55] K.-S. Choi and H. Hayashi, U(N) spectral covers from decomposition, JHEP 06 (2012) 009

[arXiv:1203.3812] [INSPIRE].

[56] J. Marsano, N. Saulina and S. Schafer-Nameki, A note on G-fluxes for F-theory model

building, JHEP 11 (2010) 088 [arXiv:1006.0483] [INSPIRE].

[57] J. Marsano, N. Saulina and S. Schafer-Nameki, On G-flux, M5 instantons and U(1)s in

F-theory, arXiv:1107.1718 [INSPIRE].

[58] T. Shioda, Mordell-Weil lattices and Galois representation. I, Proc. Japan Acad. 65 A

(1989) 268.

[59] T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39 (1990) 211.

[60] H. Hayashi, T. Kawano, R. Tatar and T. Watari, Codimension-3 singularities and Yukawa

couplings in F-theory, Nucl. Phys. B 823 (2009) 47 [arXiv:0901.4941] [INSPIRE].

[61] S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-branes and monodromy, JHEP 07

(2011) 030 [arXiv:1010.5780] [INSPIRE].

[62] T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845

(2011) 48 [arXiv:1008.4133] [INSPIRE].

[63] T.W. Grimm, D. Klevers and M. Poretschkin, Fluxes and warping for gauge couplings in

F-theory, JHEP 01 (2013) 023 [arXiv:1202.0285] [INSPIRE].

[64] R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys.

187 (1997) 679 [hep-th/9701162] [INSPIRE].

[65] P. Aluffi and M. Esole, New orientifold weak coupling limits in F-theory, JHEP 02 (2010)

020 [arXiv:0908.1572] [INSPIRE].

[66] M. Esole, J. Fullwood and S.-T. Yau, D5 elliptic fibrations: non-Kodaira fibers and new

orientifold limits of F-theory, arXiv:1110.6177 [INSPIRE].

[67] V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification

of tops, Adv. Theor. Math. Phys. 7 (2003) 205 [hep-th/0303218] [INSPIRE].

[68] C. Mayrhofer, E. Palti and T. Weigand, Models of multiple U(1) symmetries in F-theory,

work in progress.

– 44 –